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QM in Three Dimensions
The one dimensional case was good for illustrating basic features such as quantization of energy.
QM in Three Dimensions
The one dimensional case was good for illustrating basic features such as quantization of energy.
However 3-dimensions is needed for application to atomic physics, nuclear physics and other areas.
Schrödinger's Equa 3Dimensions
For 3-dimensions Schrödinger's equation becomes,
Where the Laplacian is
t
irUm
2
2
2
2
2
2
2
2
22
zyx
Schrödinger's Equa 3Dimensions
For 3-dimensions Schrödinger's equation becomes,
Where the Laplacian is
and
t
irUm
2
2
2
2
2
2
2
2
22
zyx
zyxUrU ,,
Schrödinger's Equa 3Dimensions
The stationary states are solutions to Schrödinger's equation in separable form, tiertr
,
Schrödinger's Equa 3Dimensions
The stationary states are solutions to Schrödinger's equation in separable form,
The TISE for a particle whose energy is sharp at is,
rErrUrm
22
2
tiertr ,
E
Particle in a 3 Dimensional Box
The simplest case is a particle confined to a cube of edge length L.
Particle in a 3 Dimensional Box
The simplest case is a particle confined to a cube of edge length L.
The potential energy function is
for That is, the particle is free within the box.
0,, zyxU Lzyx ,,0
Particle in a 3 Dimensional Box
The simplest case is a particle confined to a cube of edge length L.
The potential energy function is
for That is, the particle is free within the box. otherwise.
0,, zyxU Lzyx ,,0
zyxU ,,
Particle in a 3 Dimensional Box
Note: If we consider one coordinate the solution will be the same as the 1-D box.
Particle in a 3 Dimensional Box
Note: If we consider one coordinate the solution will be the same as the 1-D box.
The spatial waveform is separable (ie. can be written in product form):
r
zyxzyxr ,,
Particle in a 3 Dimensional Box
Note: If we consider one coordinate the solution will be the same as the 1-D box.
The spatial waveform is separable (ie. can be written in product form):
Substituting into the TISE and dividing by
we get,
r
zyxzyxr ,,
r
Particle in a 3 Dimensional Box
The independent variables are isolated. Each of the terms reduces to a constant:
Ezmymxm
23
2
3
2
22
2
2
2
21
2
1
2
222
121
2
1
2
2E
xm
222
2
2
2
2E
ym
323
2
3
2
2E
zm
1... 2...
3...
Particle in a 3 Dimensional Box
Clearly The solution to equations 1,2, 3 are of the
form where
EEEE 321
kxsin kxcos, 22 mEk
Particle in a 3 Dimensional Box
Clearly The solution to equations 1,2, 3 are of the
form where Applying boundary conditions we find,
EEEE 321
kxsin kxcos, 22 mEk
zkykxkAzyx 321 sinsinsin,,
Particle in a 3 Dimensional Box
Clearly The solution to equations 1,2, 3 are of the
form where Applying boundary conditions we find,
where
EEEE 321
kxsin kxcos, 22 mEk
zkykxkAzyx 321 sinsinsin,,
11 nLk 22, nLk 33, nLk
Particle in a 3 Dimensional Box
Clearly The solution to equations 1,2, 3 are of the
form where Applying boundary conditions we find,
where Therefore,
EEEE 321
kxsin kxcos, 22 mEk
zkykxkAzyx 321 sinsinsin,,
11 nLk 22, nLk 33, nLk
2322
21
2
2kkk
mE
m
pppE zyx
2
222
Particle in a 3 Dimensional Box
with and so forth. Using restrictions on the wave numbers
and boundary conditions we obtain,
1kpx L
n 1
Particle in a 3 Dimensional Box
with and so forth. Using restrictions on the wave numbers
and boundary conditions we obtain,
2322
212
22
321 2nnn
mLEEEE
1kpx L
n 1
Particle in a 3 Dimensional Box
with and so forth. Using restrictions on the wave numbers
and boundary conditions we obtain,
Thus confining a particle to a box acts to quantize its momentum and energy.
2322
212
22
321 2nnn
mLEEEE
1kpx L
n 1
Particle in a 3 Dimensional Box
Note that three quantum numbers are required to describe the quantum state of the system.
Particle in a 3 Dimensional Box
Note that three quantum numbers are required to describe the quantum state of the system.
These correspond to the three independent degrees of freedom for a particle.
Particle in a 3 Dimensional Box
Note that three quantum numbers are required to describe the quantum state of the system.
These correspond to the three independent degrees of freedom for a particle.
The quantum numbers specify values taken by the sharp observables.
Particle in a 3 Dimensional Box
The total energy will be quoted in the form
2322
212
22
,, 2321nnn
mLE nnn
Particle in a 3 Dimensional Box
Degeneracy: quantum levels (different quantum numbers) having the same energy.
Particle in a 3 Dimensional Box
Degeneracy: quantum levels (different quantum numbers) having the same energy.
Degeneracy is a natural phenomena which occurs because of the same in the system described (cubic box).
Particle in a 3 Dimensional Box
Degeneracy: quantum levels (different quantum numbers) having the same energy.
Degeneracy is a natural phenomena which occurs because of the same in the system described (cubic box).
For excited states we have degeneracy.
Particle in a 3 Dimensional Box
There are three 1st excited states having the same energy. They correspond to combinations of the quantum numbers whose squares sum to 6.
Particle in a 3 Dimensional Box
There are three 1st excited states having the same energy. They correspond to combinations of the quantum numbers whose squares sum to 6.
That is
2
22
2,1,11,2,11,1,2 2
6
mLEEE
Particle in a 3 Dimensional Box
The 1st five energy levels for a cubic box.
n2 Degeneracy
12 none
11 3
9 3
6 3
3 none
4E0
11/3E0
2E0
3E0
E0
Schrödinger's Equa 3Dimensions
The formulation in cartesian coordinates is a natural generalization from one to higher dimensions.
Schrödinger's Equa 3Dimensions
The formulation in cartesian coordinates is a natural generalization from one to higher dimensions.
However it not often best suited to a given problem. Thus it may be necessary to convert to another coordinate system.
Consider a particle in a two-dimensional (infinite) well, with Lx = Ly.
1. Compare the energies of the (2,2), (1,3), and (3,1) states? Explain your answer?
a. E(2,2) > E(1,3) = E(3,1)
b. E(2,2) = E(1,3) = E(3,1)
c. E(1,3) = E(3,1) > E(2,2)
2. If we squeeze the box in the x-direction (i.e., Lx < Ly) compare E(1,3) with E(3,1): Explain your answer?
a. E(1,3) < E(3,1)
b. E(1,3) = E(3,1)
c. E(1,3) > E(3,1)
Example 1
42
Consider a particle in a two-dimensional (infinite) well, with Lx = Ly.
1. Compare the energies of the (2,2), (1,3), and (3,1) states?
a. E(2,2) > E(1,3) = E(3,1)
b. E(2,2) = E(1,3) = E(3,1)
c. E(1,3) = E(3,1) > E(2,2)
2. If we squeeze the box in the x-direction (i.e., Lx < Ly) compare E(1,3) with E(3,1):
a. E(1,3) < E(3,1)
b. E(1,3) = E(3,1)
c. E(1,3) > E(3,1)
Example 1
E(1,3) = E(1,3) = E0 (12 + 32) = 10 E0
E(2,2) = E0 (22 + 22) = 8 E0
Example 2: Energy levels (1)• Now back to a 3D cubic box:
Show energies and label (nx,ny,nz) for the first 11 states of the particle in the 3D box, and write the degeneracy D for each allowed energy.
Use Eo= h2/8mL2.
z
x
yL
L
L
E
44
z
x
yL
L
L
D=1
6Eo (2,1,1) (1,2,1) (1,1,2)D=3
E
(1,1,1)3Eo
(nx,ny,nz)
222
2
2
8 zyxnnn nnnmL
hE
zyx
nx,ny,nz = 1,2,3,...
Example 2: Energy levels (1)• Now back to a 3D cubic box:Show energies and label (nx,ny,nz) for the first 11 states of the particle in the 3D box, and write the degeneracy D for each allowed energy.
Use Eo= h2/8mL2.
E
3Eo
6Eo
9Eo
11Eo
(nx,ny,nz)
z
x
y
L1
L2 > L1
L1
Example 3: Energy levels (2)• Now consider a non-cubic box:
22
2
222
2
1
2
88 yzxnnnn
mL
hnn
mL
hE
zyx
Assume that the box is stretched only along the y-direction. What do you think will happen to the cube’s energy levels below?
(1) The symmetry of U is “broken” for y, so the “three-fold” degeneracy is lowered…a ”two-fold” degeneracy remains due to 2 remaining equivalent directions, x and z.
(1,1,1)D=1
(1,2,1)D=1D=2(2,1,1) (1,1,2)
(2) There is an overall lowering of energies due to decreased confinement along y.
E
3Eo
6Eo
9Eo
11Eo
(nx,ny,nz)
Example 3: Energy levels (2)• Now consider a non-cubic box:
Assume that the box is stretched only along the y-direction. What do you think will happen to the cube’s energy levels below?
z
x
y
L1
L2 > L1
L1
222
222
21
2
88 yzxnnn nmL
hnn
mL
hE
zyx