Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon...
Transcript of Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon...
![Page 1: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/1.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Chapter 6 - Photons and Matter Waves
“All the fifty years ofconscious brooding havebrought me no closer toanswer the question, ‘Whatare light quanta?’ Of coursetoday every rascal thinks heknows the answer, but he isdeluding himself.”
-Albert Einstein
David J. StarlingPenn State Hazleton
PHYS 214
![Page 2: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/2.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
Light is not a classical wave of electric and
magnetic fields. Light is composed of quanta, with
energyE = hf
where h = 6.63× 10−34 J-s is the Planck constant.
How can light be both a particle and a wave?
![Page 3: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/3.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
Light is not a classical wave of electric and
magnetic fields. Light is composed of quanta, with
energyE = hf
where h = 6.63× 10−34 J-s is the Planck constant.
How can light be both a particle and a wave?
![Page 4: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/4.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
A 100 W sodium lamp emits 590 nm light. At what
rate does it emit photons?
R =Energy per Time
Energy per Photon
=Phf
=P
hc/λ
=100× 590× 10−9
6.63× 10−34 × 3× 108
= 2.97× 1020 photons/s.
![Page 5: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/5.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
A 100 W sodium lamp emits 590 nm light. At what
rate does it emit photons?
R =Energy per Time
Energy per Photon
=Phf
=P
hc/λ
=100× 590× 10−9
6.63× 10−34 × 3× 108
= 2.97× 1020 photons/s.
![Page 6: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/6.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
A 100 W sodium lamp emits 590 nm light. At what
rate does it emit photons?
R =Energy per Time
Energy per Photon
=Phf
=P
hc/λ
=100× 590× 10−9
6.63× 10−34 × 3× 108
= 2.97× 1020 photons/s.
![Page 7: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/7.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
A 100 W sodium lamp emits 590 nm light. At what
rate does it emit photons?
R =Energy per Time
Energy per Photon
=Phf
=P
hc/λ
=100× 590× 10−9
6.63× 10−34 × 3× 108
= 2.97× 1020 photons/s.
![Page 8: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/8.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
How do we know that photons have discrete
energy? Let’s set up an experiment.
First, remember: V = U/q and ∆V = ∆U/q.
Incident light is monochromatic and knocks electrons out ofthe metal.
![Page 9: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/9.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
How do we know that photons have discrete
energy? Let’s set up an experiment.
First, remember: V = U/q and ∆V = ∆U/q.
Incident light is monochromatic and knocks electrons out ofthe metal.
![Page 10: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/10.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
How do we know that photons have discrete
energy? Let’s set up an experiment.
First, remember: V = U/q and ∆V = ∆U/q.
Incident light is monochromatic and knocks electrons out ofthe metal.
![Page 11: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/11.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
How do we know that photons have discrete
energy? Let’s set up an experiment.
First, remember: V = U/q and ∆V = ∆U/q.
Incident light is monochromatic and knocks electrons out ofthe metal.
![Page 12: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/12.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
We increase the voltage until no current is
measured. This is the stopping potential Vstop.
Kmax = U = eVstop
Making the light more intense does not change the stoppingpotential.
![Page 13: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/13.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
We increase the voltage until no current is
measured. This is the stopping potential Vstop.
Kmax = U = eVstop
Making the light more intense does not change the stoppingpotential.
![Page 14: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/14.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
Alternatively, we can change the frequency of
light and measure the stopping potential for each
frequency.
There is a minimum photon frequency (energy) under whichno electrons are ejected.
![Page 15: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/15.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
Alternatively, we can change the frequency of
light and measure the stopping potential for each
frequency.
There is a minimum photon frequency (energy) under whichno electrons are ejected.
![Page 16: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/16.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
The resulting equation is the conservation of
energy: one photon energy turns into potential
and kinetic energy of an electron:
hf = Kmax + Φ
Φ is known as the “work function” and represents theenergy required to pull the electron away from the material.
![Page 17: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/17.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
The resulting equation is the conservation of
energy: one photon energy turns into potential
and kinetic energy of an electron:
hf = Kmax + Φ
Φ is known as the “work function” and represents theenergy required to pull the electron away from the material.
![Page 18: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/18.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
If we replace Kmax with eVstop, we get the another
form of the photoelectric equation.
Vstop =he
f − Φ
e
The work function depends on the material but is a constant.Therefore, we get a linear relationship.
![Page 19: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/19.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
If we replace Kmax with eVstop, we get the another
form of the photoelectric equation.
Vstop =he
f − Φ
e
The work function depends on the material but is a constant.Therefore, we get a linear relationship.
![Page 20: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/20.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Energy
Lecture Question 6.1Upon which one of the following parameters does theenergy of a photon depend?
(a) the mass of the photon
(b) the amplitude of the electric field
(c) the direction of the electric field
(d) the relative phase of the electromagnetic wave relative
to the source that produced it
(e) the frequency of the photon
![Page 21: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/21.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
Although the photon is massless, it carries
momentum p = h/λ. We know from experiments.
X-rays scatter off of a carbon target. The resulting angle,intensity and wavelength are measured.
![Page 22: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/22.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
Although the photon is massless, it carries
momentum p = h/λ. We know from experiments.
X-rays scatter off of a carbon target. The resulting angle,intensity and wavelength are measured.
![Page 23: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/23.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
Using a 71.1 pm x-ray beam, Compton measured
the following “Compton Effect.”
There is a shift in the energy of the x-ray photons as thescattering angle is changed.
![Page 24: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/24.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
Using a 71.1 pm x-ray beam, Compton measured
the following “Compton Effect.”
There is a shift in the energy of the x-ray photons as thescattering angle is changed.
![Page 25: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/25.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
The photon is colliding with an electron. Three
things can happen.
During a collision, energy must be conserved: hf = hf ′+ K.
![Page 26: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/26.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
The photon is colliding with an electron. Three
things can happen.
During a collision, energy must be conserved: hf = hf ′+ K.
![Page 27: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/27.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
After the collision, the electron has relativistic
kinetic energy: K = mc2(γ − 1),
with γ = 1/√
1− (v/c)2.
This gives:
hf = hf ′ + mc2(γ − 1)
hλ
=hλ′
+ mc(γ − 1) [I]
using c = fλ.
![Page 28: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/28.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
After the collision, the electron has relativistic
kinetic energy: K = mc2(γ − 1),
with γ = 1/√
1− (v/c)2.
This gives:
hf = hf ′ + mc2(γ − 1)
hλ
=hλ′
+ mc(γ − 1) [I]
using c = fλ.
![Page 29: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/29.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
We can also consider momentum in the x- and
y-directions.
hλ
=hλ′
cos(φ) + γmv cos(θ) [II]
0 =hλ′
sin(φ)− γmv sin(θ) [III]
![Page 30: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/30.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
We can also consider momentum in the x- and
y-directions.
hλ
=hλ′
cos(φ) + γmv cos(θ) [II]
0 =hλ′
sin(φ)− γmv sin(θ) [III]
![Page 31: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/31.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
When we combine equations I, II and III to solve
for ∆λ = λ′ − λ, we get:
∆λ =h
mc(1− cosφ)
Here, we’ve eliminated the unknown electron properties vand θ and h/mc is the Compton wavelength.
![Page 32: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/32.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Photon Momentum
When we combine equations I, II and III to solve
for ∆λ = λ′ − λ, we get:
∆λ =h
mc(1− cosφ)
Here, we’ve eliminated the unknown electron properties vand θ and h/mc is the Compton wavelength.
![Page 33: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/33.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
How do we bring together the wave nature and
particle nature of light?
During an interference experiment, we say that the photon isspread out in a probability distribution.
![Page 34: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/34.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
How do we bring together the wave nature and
particle nature of light?
During an interference experiment, we say that the photon isspread out in a probability distribution.
![Page 35: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/35.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
We associate different locations with a probability
density of detecting the photon there.
The photon is spread out like a wave while it travels, butbehaves particle-like when detected.
![Page 36: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/36.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
We associate different locations with a probability
density of detecting the photon there.
The photon is spread out like a wave while it travels, butbehaves particle-like when detected.
![Page 37: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/37.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
An interferometer can be thought of as a photon
interfering with itself.
A photon is emitted, its probability wave splits into twopaths, interferes with itself and is detected as a photon (ornot) at D.
![Page 38: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/38.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
An interferometer can be thought of as a photon
interfering with itself.
A photon is emitted, its probability wave splits into twopaths, interferes with itself and is detected as a photon (ornot) at D.
![Page 39: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/39.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
Massive particles also behave like waves, known
as matter waves, with deBroglie wavelength
λ =hp.
![Page 40: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/40.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
Massive particles also behave like waves, known
as matter waves, with deBroglie wavelength
λ =hp.
![Page 41: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/41.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
Electrons or x-rays strike the target and then
diffract like waves off of the crystalline structure.
![Page 42: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/42.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Probability Waves
Lecture Question 6.2Which one of the following experiments demonstrates thewave nature of electrons?
(a) Small flashes of light can be observed when electrons
strike a special screen.
(b) Electrons directed through a double slit can produce an
interference pattern.
(c) The Michelson-Morley experiment confirmed the
existence of electrons and their nature.
(d) In the photoelectric effect, electrons are observed to
interfere with electrons in metals.
(e) Electrons are observed to interact with photons (light
particles).
![Page 43: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/43.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
In the past, we attempted to find the exact position
~r(t) of an object using Newton’s Laws.
That is, we solve the differential equation ~Fnet = m d2~rdt2 with
initial conditions~r0 and~v0. The solution is
~r(t) = x(t)̂i + y(t)̂j + z(t)k̂
![Page 44: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/44.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
In the past, we attempted to find the exact position
~r(t) of an object using Newton’s Laws.
That is, we solve the differential equation ~Fnet = m d2~rdt2 with
initial conditions~r0 and~v0. The solution is
~r(t) = x(t)̂i + y(t)̂j + z(t)k̂
![Page 45: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/45.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
However, in quantum mechanics, particles do not
obey ~F = m~a, and exact positions~r(t) do not
exist. Instead, we have a probability density:
Ψ(x, y, z, t)
Ψ(x, y, z, t) is known as the wave function. The probabilitydensity is given by its absolute square p(x, y, z, t) = |Ψ|2.
![Page 46: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/46.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
However, in quantum mechanics, particles do not
obey ~F = m~a, and exact positions~r(t) do not
exist. Instead, we have a probability density:
Ψ(x, y, z, t)
Ψ(x, y, z, t) is known as the wave function. The probabilitydensity is given by its absolute square p(x, y, z, t) = |Ψ|2.
![Page 47: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/47.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Often, the wavefunction can by simplified:
Ψ(x, y, z, t) = ψ(x, y, z)e−iωt
where ω = 2πf is the angular frequency of the matter wave.
The probability that a detector will measure a particlebetween a position x1 and x2 is given by:
p =
∫ x2
x1
|ψ(x)|2dx→∫
V|ψ(x, y, z)|2dV.
![Page 48: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/48.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Often, the wavefunction can by simplified:
Ψ(x, y, z, t) = ψ(x, y, z)e−iωt
where ω = 2πf is the angular frequency of the matter wave.
The probability that a detector will measure a particlebetween a position x1 and x2 is given by:
p =
∫ x2
x1
|ψ(x)|2dx→∫
V|ψ(x, y, z)|2dV.
![Page 49: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/49.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This wavefunction Ψ, like~r, is the solution to an
important differential equation.
Schrödinger’s Equation!
E = K + U
=12
mv2 + U
=1
2m(mv)2 + U
=p2
2m+ U
Eψ =p2
2mψ + Uψ.
![Page 50: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/50.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This wavefunction Ψ, like~r, is the solution to an
important differential equation.
Schrödinger’s Equation!
E = K + U
=12
mv2 + U
=1
2m(mv)2 + U
=p2
2m+ U
Eψ =p2
2mψ + Uψ.
![Page 51: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/51.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This wavefunction Ψ, like~r, is the solution to an
important differential equation.
Schrödinger’s Equation!
E = K + U
=12
mv2 + U
=1
2m(mv)2 + U
=p2
2m+ U
Eψ =p2
2mψ + Uψ.
![Page 52: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/52.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This wavefunction Ψ, like~r, is the solution to an
important differential equation.
Schrödinger’s Equation!
E = K + U
=12
mv2 + U
=1
2m(mv)2 + U
=p2
2m+ U
Eψ =p2
2mψ + Uψ.
![Page 53: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/53.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This wavefunction Ψ, like~r, is the solution to an
important differential equation.
Schrödinger’s Equation!
E = K + U
=12
mv2 + U
=1
2m(mv)2 + U
=p2
2m+ U
Eψ =p2
2mψ + Uψ.
![Page 54: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/54.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This wavefunction Ψ, like~r, is the solution to an
important differential equation.
Schrödinger’s Equation!
E = K + U
=12
mv2 + U
=1
2m(mv)2 + U
=p2
2m+ U
Eψ =p2
2mψ + Uψ.
![Page 55: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/55.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This wavefunction Ψ, like~r, is the solution to an
important differential equation.
Schrödinger’s Equation!
E = K + U
=12
mv2 + U
=1
2m(mv)2 + U
=p2
2m+ U
Eψ =p2
2mψ + Uψ.
![Page 56: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/56.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s assume that the wavefunction, which
describes our wave-like electrons, is oscillatory.
ψ(x, t) ∝ ei(kx−ωt)
with k = 2π/λ = p/~ and ~ = h/2π.
d2ψ
dx2 = (ik)2ψ
d2ψ
dx2 = −p2
~2ψ
− ~2 d2ψ
dx2 = p2ψ
− ~2
2md2ψ
dx2 =p2
2mψ
![Page 57: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/57.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s assume that the wavefunction, which
describes our wave-like electrons, is oscillatory.
ψ(x, t) ∝ ei(kx−ωt)
with k = 2π/λ = p/~ and ~ = h/2π.
d2ψ
dx2 = (ik)2ψ
d2ψ
dx2 = −p2
~2ψ
− ~2 d2ψ
dx2 = p2ψ
− ~2
2md2ψ
dx2 =p2
2mψ
![Page 58: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/58.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s assume that the wavefunction, which
describes our wave-like electrons, is oscillatory.
ψ(x, t) ∝ ei(kx−ωt)
with k = 2π/λ = p/~ and ~ = h/2π.
d2ψ
dx2 = (ik)2ψ
d2ψ
dx2 = −p2
~2ψ
− ~2 d2ψ
dx2 = p2ψ
− ~2
2md2ψ
dx2 =p2
2mψ
![Page 59: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/59.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Putting this all together, we get the
time-independent Schrödinger’s equation.
− ~2
2md2ψ
dx2 + Uψ = Eψ
or,d2ψ
dx2 +2m~2 (E − U)ψ = 0
Caveats:
(a) This is only 1-dimensional
(b) This ignores the time-oscillation
(c) The solution depends on the function U(x)
![Page 60: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/60.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Putting this all together, we get the
time-independent Schrödinger’s equation.
− ~2
2md2ψ
dx2 + Uψ = Eψ
or,d2ψ
dx2 +2m~2 (E − U)ψ = 0
Caveats:
(a) This is only 1-dimensional
(b) This ignores the time-oscillation
(c) The solution depends on the function U(x)
![Page 61: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/61.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s solve Schrödinger’s Equation for a free
particle (i.e., U(x) = 0).
d2ψ
dx2 +2m~2 (E)ψ = 0
d2ψ
dx2 +2m~2 (
12
mv2)ψ = 0
d2ψ
dx2 +(mv)2
~2 ψ = 0
d2ψ
dx2 +(p~
)2ψ = 0
d2ψ
dx2 + k2ψ = 0
The general solution is:
ψ(x) = Aeikx + Be−ikx → Aeikx(right traveling)
![Page 62: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/62.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s solve Schrödinger’s Equation for a free
particle (i.e., U(x) = 0).
d2ψ
dx2 +2m~2 (E)ψ = 0
d2ψ
dx2 +2m~2 (
12
mv2)ψ = 0
d2ψ
dx2 +(mv)2
~2 ψ = 0
d2ψ
dx2 +(p~
)2ψ = 0
d2ψ
dx2 + k2ψ = 0
The general solution is:
ψ(x) = Aeikx + Be−ikx → Aeikx(right traveling)
![Page 63: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/63.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s solve Schrödinger’s Equation for a free
particle (i.e., U(x) = 0).
d2ψ
dx2 +2m~2 (E)ψ = 0
d2ψ
dx2 +2m~2 (
12
mv2)ψ = 0
d2ψ
dx2 +(mv)2
~2 ψ = 0
d2ψ
dx2 +(p~
)2ψ = 0
d2ψ
dx2 + k2ψ = 0
The general solution is:
ψ(x) = Aeikx + Be−ikx → Aeikx(right traveling)
![Page 64: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/64.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s solve Schrödinger’s Equation for a free
particle (i.e., U(x) = 0).
d2ψ
dx2 +2m~2 (E)ψ = 0
d2ψ
dx2 +2m~2 (
12
mv2)ψ = 0
d2ψ
dx2 +(mv)2
~2 ψ = 0
d2ψ
dx2 +(p~
)2ψ = 0
d2ψ
dx2 + k2ψ = 0
The general solution is:
ψ(x) = Aeikx + Be−ikx → Aeikx(right traveling)
![Page 65: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/65.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s solve Schrödinger’s Equation for a free
particle (i.e., U(x) = 0).
d2ψ
dx2 +2m~2 (E)ψ = 0
d2ψ
dx2 +2m~2 (
12
mv2)ψ = 0
d2ψ
dx2 +(mv)2
~2 ψ = 0
d2ψ
dx2 +(p~
)2ψ = 0
d2ψ
dx2 + k2ψ = 0
The general solution is:
ψ(x) = Aeikx + Be−ikx → Aeikx(right traveling)
![Page 66: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/66.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s solve Schrödinger’s Equation for a free
particle (i.e., U(x) = 0).
d2ψ
dx2 +2m~2 (E)ψ = 0
d2ψ
dx2 +2m~2 (
12
mv2)ψ = 0
d2ψ
dx2 +(mv)2
~2 ψ = 0
d2ψ
dx2 +(p~
)2ψ = 0
d2ψ
dx2 + k2ψ = 0
The general solution is:
ψ(x) = Aeikx + Be−ikx → Aeikx(right traveling)
![Page 67: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/67.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Let’s solve Schrödinger’s Equation for a free
particle (i.e., U(x) = 0).
d2ψ
dx2 +2m~2 (E)ψ = 0
d2ψ
dx2 +2m~2 (
12
mv2)ψ = 0
d2ψ
dx2 +(mv)2
~2 ψ = 0
d2ψ
dx2 +(p~
)2ψ = 0
d2ψ
dx2 + k2ψ = 0
The general solution is:
ψ(x) = Aeikx + Be−ikx → Aeikx(right traveling)
![Page 68: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/68.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This solution tells us how a free particle moves in
1D. The probability density is the absolute square.
p = |ψ(x)|2
= ψ∗(x)ψ(x)
= (Ae−ikx)× (Aeikx)
= A2
= constant
![Page 69: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/69.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This solution tells us how a free particle moves in
1D. The probability density is the absolute square.
p = |ψ(x)|2
= ψ∗(x)ψ(x)
= (Ae−ikx)× (Aeikx)
= A2
= constant
![Page 70: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/70.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This solution tells us how a free particle moves in
1D. The probability density is the absolute square.
p = |ψ(x)|2
= ψ∗(x)ψ(x)
= (Ae−ikx)× (Aeikx)
= A2
= constant
![Page 71: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/71.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This solution tells us how a free particle moves in
1D. The probability density is the absolute square.
p = |ψ(x)|2
= ψ∗(x)ψ(x)
= (Ae−ikx)× (Aeikx)
= A2
= constant
![Page 72: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/72.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
This solution tells us how a free particle moves in
1D. The probability density is the absolute square.
p = |ψ(x)|2
= ψ∗(x)ψ(x)
= (Ae−ikx)× (Aeikx)
= A2
= constant
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Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
One consequence of this probabilistic behavior is
the Heisenberg Uncertainty Principle.
∆x ·∆px ≥ ~
The less uncertainty in position, the more uncertainty inmomentum (and vice versa).
![Page 74: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/74.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
One consequence of this probabilistic behavior is
the Heisenberg Uncertainty Principle.
∆x ·∆px ≥ ~
The less uncertainty in position, the more uncertainty inmomentum (and vice versa).
![Page 75: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/75.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Schrödinger’s Equation
Lecture Question 6.3Which one of the following statements provides the bestdescription of the Heisenberg Uncertainty Principle?
(a) If a particle is confined to a region ∆x, then its
momentum is within some range ∆p.
(b) If the error in measuring the position is ∆x, then we
can determine the error in measuring the momentum
∆p.
(c) If one measures the position of a particle, then the
value of the momentum will change.
(d) It is not possible to be certain of any measurement.
(e) Depending on the degree of certainty in measuring the
position of a particle, the degree of certainty in
measuring the momentum is affected.
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Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
Consider a puck sliding along an icy hill where
Ub = mgh is the potential energy at the top.
In this case, the puck needs K > Ub to pass over this barrier.
![Page 77: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/77.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
Consider a puck sliding along an icy hill where
Ub = mgh is the potential energy at the top.
In this case, the puck needs K > Ub to pass over this barrier.
![Page 78: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/78.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
However, in quantum mechanics, a particle can
tunnel through a barrier even if it does not have
enough energy to do so.
We must solve Schrödinger’s equation in order to find thetunneling probability!
![Page 79: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/79.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
However, in quantum mechanics, a particle can
tunnel through a barrier even if it does not have
enough energy to do so.
We must solve Schrödinger’s equation in order to find thetunneling probability!
![Page 80: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/80.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
d2ψ
dx2 +2m~2 (E − U)ψ = 0
I For x < 0 and x > L, we have a free particle U = 0.
I For 0 < x < L, E < Ub = eVb.
I At each boundary, ψ(x) must be continuous and
smooth.
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Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
d2ψ
dx2 +2m~2 (E − U)ψ = 0
I For x < 0 and x > L, we have a free particle U = 0.
I For 0 < x < L, E < Ub = eVb.
I At each boundary, ψ(x) must be continuous and
smooth.
![Page 82: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/82.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
The term E − U is the kinetic energy.
E − U =12
mv2 =p2
2m=
12m
(k~)2
(note: p = h/λ = k~)
Substituting in:
d2ψ
dx2 + k2ψ = 0
I This has the same solutions as before (eikx)
I Here, k =√
2m(E − U)~
I The wavenumber can be imaginary if U > E.
![Page 83: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/83.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
The term E − U is the kinetic energy.
E − U =12
mv2 =p2
2m=
12m
(k~)2
(note: p = h/λ = k~) Substituting in:
d2ψ
dx2 + k2ψ = 0
I This has the same solutions as before (eikx)
I Here, k =√
2m(E − U)~
I The wavenumber can be imaginary if U > E.
![Page 84: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/84.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
The term E − U is the kinetic energy.
E − U =12
mv2 =p2
2m=
12m
(k~)2
(note: p = h/λ = k~) Substituting in:
d2ψ
dx2 + k2ψ = 0
I This has the same solutions as before (eikx)
I Here, k =√
2m(E − U)~
I The wavenumber can be imaginary if U > E.
![Page 85: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/85.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
There are three regions when an electron hits a
barrier:
I Left Side (k ∈ R): ψ(x) = Aeikx + Be−ikx
I Middle Section (k ∈ I): ψ(x) = Ceikx + De−ikx
I Right Side (k ∈ R): ψ(x) = Eeikx
I At each boundary, ψ(x) must be continuous and
smooth.
![Page 86: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/86.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
There are three regions when an electron hits a
barrier:
I Left Side (k ∈ R): ψ(x) = Aeikx + Be−ikx
I Middle Section (k ∈ I): ψ(x) = Ceikx + De−ikx
I Right Side (k ∈ R): ψ(x) = Eeikx
I At each boundary, ψ(x) must be continuous and
smooth.
![Page 87: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/87.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
There are three regions when an electron hits a
barrier:
I Left Side (k ∈ R): ψ(x) = Aeikx + Be−ikx
I Middle Section (k ∈ I): ψ(x) = Ceikx + De−ikx
I Right Side (k ∈ R): ψ(x) = Eeikx
I At each boundary, ψ(x) must be continuous and
smooth.
![Page 88: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/88.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
There are three regions when an electron hits a
barrier:
I Left Side (k ∈ R): ψ(x) = Aeikx + Be−ikx
I Middle Section (k ∈ I): ψ(x) = Ceikx + De−ikx
I Right Side (k ∈ R): ψ(x) = Eeikx
I At each boundary, ψ(x) must be continuous and
smooth.
![Page 89: Chapter 6 - Photons and Matter Waves · Chapter 6 - Photons and Matter Waves Photon Energy Photon Momentum Probability Waves Schrödinger’s Equation Tunneling Photon Energy ...](https://reader034.fdocuments.in/reader034/viewer/2022042711/5f6cee04a5aa306418596ac9/html5/thumbnails/89.jpg)
Chapter 6 - Photons andMatter Waves
Photon Energy
Photon Momentum
Probability Waves
Schrödinger’s Equation
Tunneling
Tunneling
There are three regions when an electron hits a
barrier:
I Left Side (k ∈ R): ψ(x) = Aeikx + Be−ikx
I Middle Section (k ∈ I): ψ(x) = Ceikx + De−ikx
I Right Side (k ∈ R): ψ(x) = Eeikx
I At each boundary, ψ(x) must be continuous and
smooth.