3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides:...
Transcript of 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides:...
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3.2 Finite well and harmonic oscillator
Slides: Video 3.2.1 Particles in potential wells – introduction
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Particles in potential wells
David MillerQuantum mechanics for scientists and engineers
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3.2 Finite well and harmonic oscillator
Slides: Video 3.2.2 The finite potential well
Text reference: Quantum Mechanics for Scientists and Engineers
Section 2.9
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Particles in potential wells
The finite potential well
Quantum mechanics for scientists and engineers David Miller
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Finite potential wellInsert video here (split
screen)
Lesson 7Particles in potential
wells
Insert number 2
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Particle in a finite potential well
We will choose the height of the potential barriers as Vowith 0 potential energy at the
bottom of the wellThe thickness of the well is Lz
Now we will choose the position origin in the center of the well
oV
00/ 2zL / 2zL z
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Particle in a finite potential well
If there is an eigenenergy E for which there is a solutionthen we already know what
form the solution has to takesinusoidal in the middleexponentially decaying on either side
oV
00/ 2zL / 2zL z
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Particle in a finite potential well
For some eigenenergy Ewith and
for
for
for
with constants A, B, F, and G
oV
00/ 2zL / 2zL z
/ 2zz L expz G z
/ 2 / 2z zL z L sin cosz A kz B kz
/ 2zz L expz F z
22 /k mE 22 /om V E
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Particle in a finite potential well
Now we need to apply the boundary conditions to solve for the unknown coefficientsconstants A, B, F, and G
or at least three of themthe fourth could be found by normalization
oV
00/ 2zL / 2zL z
/ 2zz L expz G z
/ 2 / 2z zL z L sin cosz A kz B kz / 2zz L expz F z
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Particle in a finite potential well
From continuity of the wavefunction at
Writing
gives
oV
00/ 2zL / 2zL z
/ 2 exp / 2z zL F L
/ 2zz L
exp / 2L zX L
sin / 2L zS kL cos / 2L zC kL
L L LFX AS BC
sin / 2 cos / 2z zA kL B kL
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Particle in a finite potential well
Similarly at
Continuity of the derivative givesat
at
oV
00/ 2zL / 2zL z
/ 2zz L
L L LGX AS BC
L L LGX AC BSk
/ 2zz L
/ 2zz L
L L LFX AC BSk
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Particle in a finite potential well
So we have four relations
Now we need to find what solutions are compatible with these
oV
00/ 2zL / 2zL z
L L LGX AS BC
L L LGX AC BSk
L L LFX AC BSk
L L LFX AS BC
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Particle in a finite potential well
Adding
gives
Subtracting
from
gives
oV
00/ 2zL / 2zL z
L L LGX AS BC
L L LFX AS BC
2 L LBC F G X
L L LFX AC BSk
L L LGX AC BSk
2 L LBS F G Xk
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Particle in a finite potential well
As long aswe can divide
by
to obtain
This relation is effectively a condition for eigenvalues
oV
00/ 2zL / 2zL z
2 L LBC F G X
2 L LBS F G Xk
F G
tan / 2 /zkL k
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Particle in a finite potential well
Subtractingfrom gives
Adding
and
gives
oV
00/ 2zL / 2zL z
L L LGX AS BC
L L LFX AS BC
L L LFX AC BSk
L L LGX AC BSk
2 L LAS F G X
2 L LAC F G Xk
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Particle in a finite potential well
Similarly, as long aswe can divide
by
to obtain
This relation is also effectively a condition for eigenvalues
oV
00/ 2zL / 2zL z
F G
2 L LAS F G X
2 L LAC F G Xk
cot / 2 /zkL k
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Particle in a finite potential well
For any case other thanwhich leaves
but notor
which leavesbut not
then the solutions and are contradictory
oV
00/ 2zL / 2zL z
F G
cot / 2 /zkL k tan / 2 /zkL k
F G cot / 2 /zkL k
tan / 2 /zkL k
tan / 2 /zkL k cot / 2 /zkL k
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Particle in a finite potential well
So the only possibilities are
1 -and
2 –and
oV
00/ 2zL / 2zL z
F G cot / 2 /zkL k
F G tan / 2 /zkL k
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Particle in a finite potential well
1 -and
Note from
and
SL and CL cannot both be 0so
Hence in the well we have
which is an even function
oV
00/ 2zL / 2zL z
F G tan / 2 /zkL k
2 L LAS F G X
2 L LAC F G Xk
0A
cosz kz
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Particle in a finite potential well
1 -and
Note from
and
SL and CL cannot both be 0so
Hence in the well we have
which is an odd function
oV
00/ 2zL / 2zL z
0B
sinz kz
F G cot / 2 /zkL k
2 L LBS F G Xk
2 L LBC F G X
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Particle in a finite potential well
Though we have found the nature of the solutionswe have not yet formally
solved for the eigenenergiesEand hence for k and
We do this by solving
and
oV
00/ 2zL / 2zL z
tan / 2 /zkL k
cot / 2 /zkL k
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Solving for the eigenenergies
Change to “dimensionless” unitsUse the energy of the first level in the
“infinite” potential well width Lzleading to a dimensionless
eigenenergyand a dimensionless barrier height
Also
22
1 2 z
Em L
1/E E
1/o ov V E
212 / / / /z zk mE L E E L
212 / / / /o z o z om V E L V E E L
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Consequently
and
So becomes
or
and becomes
or cot / 2 ov
tan / 2 ov
Solving for the eigenenergies
o oV E vk E
12 2 2zkL E
E
12 2 2oz
oV EL v
E
tan / 2 /zkL k tan / 2 /ov
cot / 2 /zkL k cot / 2 /ov
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Graphical solution
Choose a specific well depth and plot the curve
ov
2 4 6 8
-2
0
2
4
6
2o
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Graphical solution
Choose a specific well depth and plot the curve
2 4 6 8
-2
0
2
4
6
5o
ov
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Graphical solution
Choose a specific well depth and plot the curve
2 4 6 8
-2
0
2
4
6
8o
ov
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Graphical solution
Choose a specific well depth and plot the curve
Now add the curves
2 4 6 8
-2
0
2
4
6
8o
ov
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Graphical solution
Choose a specific well depth and plot the curve
Now add the curves
2 4 6 8
-2
0
2
4
6
8o
tan2
ov
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Graphical solution
Choose a specific well depth and plot the curve
Now add the curves
2 4 6 8
-2
0
2
4
6
8o
cot2
tan
2
ov
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Graphical solution
For a specific the solutions are the values
of at the intersections of
and
or
ov
2 4 6 8
-2
0
2
4
6
8o
0.663 2.603 5.609
tan2
cot2
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Solutions
These are the solutions for a well depth Vo ofNote that
they are all lower energies
than the corresponding solutions for the infinitely deep well of the same width
18E
1n
2n
3n
18oV E
00.663
2.603
5.609
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3.2 Finite well and harmonic oscillator
Slides: Video 3.2.4 The harmonic oscillator
Text reference: Quantum Mechanics for Scientists and Engineers
Section 2.10
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Particles in potential wells
The harmonic oscillator
Quantum mechanics for scientists and engineers David Miller
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Mass on a spring
A simple spring will have a restoring force F acting on the mass M
S
Mass M
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S
Mass on a spring
A simple spring will have a restoring force F acting on the mass Mproportional to the amount y by which
it is stretchedFor some “spring constant” K
The minus sign is because this is “restoring”it is trying to pull y back towards zero
This gives a “simple harmonic oscillator”
F Ky
y
Mass MForce F
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Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g., S
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
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S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 40: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/40.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 41: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/41.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 42: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/42.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 43: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/43.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 44: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/44.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 45: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/45.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 46: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/46.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 47: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/47.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 48: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/48.jpg)
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g., S
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 49: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/49.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 50: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/50.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 51: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/51.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 52: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/52.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 53: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/53.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 54: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/54.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 55: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/55.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 56: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/56.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 57: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/57.jpg)
S
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g.,
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 58: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/58.jpg)
Mass on a spring
From Newton’s second law
i.e.,
where we definewe have oscillatory solutions of angular frequencye.g., S
2
2
d yF Ma M Kydt
2
22
d y K y ydt M
2 /K M
/K M
siny t
![Page 59: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/59.jpg)
Potential energy
The potential from the restoring force F is S
z
Mass mForce F
0
z
oV z F dz
212
V z Kz
z
0
z
o oKz dz 212
Kz 2 212
m z
![Page 60: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/60.jpg)
Harmonic oscillator Schrödinger equation
With this potential energythe Schrödinger equation is
For convenience, we define a dimensionless distance unit
so the Schrödinger equation becomes
2 212
V z m z
2 22 2
2
12 2
d m z Em dz
m z
2 2
2
12 2
d Ed
![Page 61: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/61.jpg)
Harmonic oscillator Schrödinger equation
One specific solution to this equation
is
with a corresponding energy This suggests we look for solutions of the form
where is some set of functions still to be determined
2exp( / 2)
/ 2E
2exp / 2n n nA H
nH
2 2
2
12 2
d Ed
![Page 62: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/62.jpg)
Harmonic oscillator Schrödinger equation
Substitutinginto the Schrödinger equation
gives
This is the defining differential equation for the Hermite polynomials
2exp / 2n n nA H
2
2
22 1 0n nn
d H dH E Hd d
2 2
2
12 2
d Ed
![Page 63: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/63.jpg)
Harmonic oscillator Schrödinger equation
Solutions to
exist provided
that is,
2
2
22 1 0n nn
d H dH E Hd d
2 1 2E n
0, 1, 2,n
12nE n
0, 1, 2,n
![Page 64: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/64.jpg)
Harmonic oscillator Schrödinger equation
The allowed energy levels are equally spaced separated by an amount
where is the classical oscillation frequency Like the potential well
there is a “zero point energy” here
/ 2
12nE n
0, 1, 2,n
![Page 65: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/65.jpg)
Hermite polynomials
The first Hermite polynomials areNote they are either
odd or eveni.e., they have a definite parity
They satisfy a “recurrence relation”
successive Hermite polynomialscan be calculated from the previous two
0 1H
1 2H
22 4 2H
33 8 12H
4 24 16 48 12H 1 22 2 1n n nH H n H
![Page 66: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/66.jpg)
12nE n
0, 1, 2,n
m z
Harmonic oscillator solutions
Normalizing
gives
2exp / 2n n nA H
12 !n n
An
![Page 67: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/67.jpg)
Harmonic oscillator solutions
Normalizing
gives
or
2exp / 2n n nA H
12 !n n
An
21 exp2 ! 2n nn
m m mz z H zn
12nE n
0, 1, 2,n
m z
![Page 68: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/68.jpg)
Harmonic oscillator eigensolutions
4 2 0 2 4
2
4
20 exp / 2A
/ 2
21 exp / 2 2A
3 / 2
2 22 exp / 2 4 2A
5 / 2
2 33 exp / 2 8 12A
7 / 2
2 4 24 exp / 2 16 48 12A
9 / 2Energy 2 / 2
![Page 69: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/69.jpg)
Classical turning points
The intersections of the parabola
and the dashed lines
give the “classical turning points”
where a classical mass of that energy turns round and goes back downhill
4 2 0 2 4
2
4
/ 2
3 / 2
5 / 2
7 / 2
9 / 2Energy 2 / 2
![Page 70: 3.2 Finite well and harmonic oscillator - | Stanford … Finite well and harmonic oscillator Slides: Video 3.2.4 The harmonic oscillator Text reference: Quantum Mechanics for Scientists](https://reader031.fdocuments.in/reader031/viewer/2022021513/5b08e64a7f8b9a520e8d5cbc/html5/thumbnails/70.jpg)