Quantum Physics IIQuantum Physics II

UNCERTAINTY UNCERTAINTY PRINCIPLESPRINCIPLES

What are uncertainty What are uncertainty principles?principles?In QM, the product of uncertainties

in variables is non-zero◦Position-momentum◦Energy-time

Intrinsic imprecision, not due to measurement limits

Measurements (e.g. px and x) on identical systems do not yield consistent results

x

hp x

4h

E t

4

Is vacuum really empty?Is vacuum really empty?Energy can be “borrowed” from

nothing on the condition that this energy is returned within a certain time governed by the energy-time uncertainty principle◦This borrowed energy becomes the

mass of particles (E = mc2)◦The larger the energy “borrowed”, the

shorter its lifetime The larger the mass of the particle created,

the shorter is its lifetime

WHAT IS WAVE WHAT IS WAVE FUNCTION?FUNCTION?

Pg 5 – 7

Definition*Definition*Wave function of a particle is a

complex quantity that is the probability amplitude of which the absolute square gives the probability density function for locating the particle within regions of space

The rest of this section is not in your syllabus! So don’t panic if you don’t understand at all

What is wave function?What is wave function?Recap: Wave experiments

Single slit experiment

Double- slit experiment

What is wave function?What is wave function?

Refer to Dr. Quantum movieclip http://www.youtube.com/watch?v=DfPeprQ7oGc

What is wave function?What is wave function? If e- are used

instead of lightPattern builds up

spot by spot◦ Particle-like

Distribution is interference pattern!◦ Wave-like

Intensity represents probability of electron landing on that region

What is wave function?What is wave function?Let’s return to interference of

wavesThe intensity at a region is

Using principle of superposition

The intensity at any constructive interference region is

Intensity, I kx02

Net displacement,xR x1 x2

IR kxR2 k x01 x02 2

What is wave function?What is wave function?Since e- beams thru 2 slits

interfere and the intensity at a region represents probability

Rewriting the interference relation for e-

What passes thru each slit must be the probability amplitude!

Probability 1 2

2

Iwave x01 x02 2

What is wave function?What is wave function?Wave function is a complex (not a

pun) quantity interpreted as the probability amplitude

The absolute square gives the probability density function

Probability of finding particle between x1 and x2 is

Prob. density function2

Iwave x02

P x1 x x2 2

x1

x2 dx

What is wave function?What is wave function?Wave treatment of particle is

basically statistical◦Conventional statistics

◦Quantum mechanically

Do you notice the similarity?◦Hence the absolute square of wave

function gives the probability density function

x xf x dx

x2 x2f x dx

x xdx

x2 x2dx

Electrons through double Electrons through double slitsslits

Assume that the particle (eg electron) can be represented by a mathematical expression

eg a wave function (which could be complex) and also assume that the intensity profile of the interference pattern (eg the numbers of electrons detected per second) can be expressed by the square of the absolute value of this

wave function | |2

If slit 1 is opened (slit 2 closed), then we can represent the wave function of the electrons passing through slit 1 as 1 and therefore the intensity profile is | 1 |2

If slit 2 is opened (slit 1 closed), then we can represent the wave function of the electrons passing through slit 1 as 2 and therefore the intensity profile is | 2 |2

If we open slit 1 for half the time (slit 2 closed) and then slit 2 for half the time (slit 1 closed) then the intensity profile will be | 1 |2 + | 2 |2

If we open both slits then the electron wave functions are superimposed

(similar to light). The combined wave function is then 1 + 2

The intensity profile is then | 1 + 2 | 2 =

| 1 |2 + | 2 |2 + 2 (1 . 2)

This is different from the previous case of opening one slit 50% of the time, ie | 1 |2 + | 2 |2

The term 2 ( 1 . 2 ) represents the interference term.

Note that if the wave functions are complex, then | 1 |2 = 1 1* (where 1* is the complex conjugate)

Where have you seen wave Where have you seen wave functions?functions?Electron clouds and orbitals in

Chemistry◦Orbitals are square of wave

functions!

In quantum mechanics 2 is proportional to the probability of finding the particle at a given location.

The probability density |The probability density ||2

||2

QUANTUM QUANTUM TUNNELLINGTUNNELLING

Pg 8 – 9

Potential barrierPotential barrierConsider the GPE of a mass m

near Earth’s surface

GPEGPE = mgh

h

Potential barrierPotential barrierIf a particle have total energy ET

is projected upwards from the ground (GPE define as 0)

GPEGPE = mgh

h

EP = 0 Ek = ET

EP > 0 Ek = ET - EP

EP = ET Ek = 0

ET

EK

Turn back at this height

Potential barrierPotential barrierConsider an arbitrary PE for a

mass mPE

h

Potential barrierPotential barrierIf the mass m has total energy ET,

is projected from h = 0 with EKPE

h

EP = 0 Ek = ET

EP = ET Ek = 0Turn back!

ET

EK

h0

Potential barrierPotential barrierThe potential confines the

particle within a region, it is not allowed beyond h0 PE

hh0

ET

EK

Potential barrierPotential barrierPotential barrier are

gravitational, electrical in nature◦Related to potential energy◦Invisible, not a physical obstacle!

It is a barrier when the potential energy of the particle at a particular position(s) in space is larger than the particle’s energy◦ie, the particle cannot reach such

position(s) given its current total energy

Quantum tunnellingQuantum tunnellingClassically particle cannot move

into and past the region of the potential barrier because its energy is not sufficient

Quantum tunnellingQuantum tunnellingThe wave treatment of particle

allows a finite probability in/beyond the region of the potential barrier

reflected transmitted

T exp 2kL where k 2m U E

hR T 1

Wave function in potential Wave function in potential barrierbarrierSome examples of wave

functions in well-known potential

APPLICATION OF APPLICATION OF TUNNELLINGTUNNELLING

Pg 10 – 12

Scanning tunnelling Scanning tunnelling microscopemicroscope

Scanning tunnelling Scanning tunnelling microscopemicroscopePotential barrier is

the gapTunnelling when the

gap is small enough◦Tunnelling current

Small applied p.d. for a fix current direction

Refer to Eg 9 for modes of operation

Alpha decayAlpha decay

A-LEVEL QUESTIONSA-LEVEL QUESTIONS

Q1 – SP07/III/8dQ1 – SP07/III/8dA electron in an atom may

be considered to be a potential well, as illustrated by the sketch graph

Explain how, by considering the wave function of the electron, rather than by considering it as a particle, there is a possibility of the electron escaping from the potential well by a process called tunnelling.

Distance from centre of atom

energy level of electron in atom

Q1 – SolutionQ1 – Solution

Classically, an electron could never exist outside the potential barrier imposed by the atom because it does not have sufficient energy

If the electron is treated as a wave and applying Schrodinger equation, its wave function◦ is sinusoidal with large amplitude between the

barrier◦ decays exponentially within the barrier◦ is sinusoidal with a much smaller amplitude outside

the atomThe square of the wave function gives a

small but finite probability of finding the electron outside the atom

Barrier width

Q2 – SP07/III/8eQ2 – SP07/III/8eThe process in Q1 is used in a

scanning tunnelling microscope, where magnifications of up to 108 make it possible to see individual atoms. Outline how these atomic-scale images may be obtained.

Q3 - N07/III/7eQ3 - N07/III/7eShow, with the aid of a diagram,

what is meant by a potential barrier. Discuss how the wave nature of particles allows particles to tunnel through such a barrier.

Q3 – SolutionQ3 – Solution

Classically, an electron could never exist on the right of the potential barrier because it does not have sufficient energy If the electron is treated as a wave and applying Schrodinger equation, its wave function◦ is sinusoidal with large amplitude before the barrier◦ decays exponentially within the barrier◦ is sinusoidal with a much smaller amplitude after

the barrierThe square of the wave function gives a

small but finite probability of finding the electron to the right of the barrier

Energy of electron

PE of electron

x1 x2 x

EXTRA QUESTIONSEXTRA QUESTIONS

H1H1What is the uncertainty in the

location of a photon of wavelength 300 nm if this wavelength is known to an accuracy of one part in a million?

[23.9 mm]

H2H2If we assume that the energy of a

particle moving in a straight line to be mv2/2, show that the energy-time uncertainty principle is given by h

E t

4

H3H3The width of a spectral line of

wavelength 400 nm is measured to be 10-14 m. What is the average time the atomic system remains in the corresponding energy state?

[4.24 x 10-9 s]

H4H4A particle of mass m is confined to

a one-dimensional line of length L◦Find the expression of the smallest

energy that the body can have◦What is the significance of this value?◦Calculate the minimum KE, in eV, of a

neutron in a nucleus of diameter 10-14 m

[0.013 MeV]

H5*H5*If the energy width of an excited

state of a system is 1.1 eV and its excitation energy is 1.6 keV,◦what is the the average lifetime of

that state?◦what is the minimum uncertainty in

the wavelength of the photon emitted when the system de-excites?

[2.99 x 10-16 s, 5.33 x 10-13 m]