Schrodinger’s Equation for Three Dimensions

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,

Schrödinger's Equa 3Dimensions For 3-dimensions Schrödinger's equation becomes, Where the Laplacian is

Schrödinger's Equa 3Dimensions For 3-dimensions Schrödinger's equation becomes, Where the Laplacian is and Schrödinger's equation in 3D is made up of the 1D equations for the independent axes. The equations have the same form!

Schrödinger's Equa 3Dimensions The stationary states are solutions to Schrödinger's equation in separable form,

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,

Particle in a 3 Dimensional Box

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

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.

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.

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):

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,

Particle in a 3 Dimensional Box The independent variables are isolated. Each of the terms reduces to a constant:

Particle in a 3 Dimensional Box Clearly

Particle in a 3 Dimensional Box Clearly The solution to equations 1,2, 3 are of the form where

Particle in a 3 Dimensional Box Clearly The solution to equations 1,2, 3 are of the form where Applying boundary conditions we find,

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

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,

Particle in a 3 Dimensional Box with and so forth.

Particle in a 3 Dimensional Box with and so forth. Using restrictions on the wave numbers and boundary conditions we obtain,

Particle in a 3 Dimensional Box with and so forth. Using restrictions on the wave numbers and boundary conditions we obtain,

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.

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

Particle in a 3 Dimensional Box The ground state ( ) has energy

Particle in a 3 Dimensional Box Degeneracy

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

Particle in a 3 Dimensional Box The 1st five energy levels for a cubic box. n2 Degeneracy 12 none 11 3 9 6 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.

Schrödinger's Equa 3Dimensions Consider an electron orbiting a central nucleus.

Example 1 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 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) E(1,3) = E(1,3) = E0 (12 + 32) = 10 E0 E(2,2) = E0 (22 + 22) = 8 E0

Example 2: Energy levels (1) z x y L 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

Example 2: Energy levels (1) z x y L 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. (1,1,1) 3Eo (nx,ny,nz) E nx,ny,nz = 1,2,3,... 6Eo (2,1,1) (1,2,1) (1,1,2) D=3 D=1

Example 3: Energy levels (2) z x y L1 L2 > L1 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? E (nx,ny,nz) 11Eo 9Eo 6Eo 3Eo

Example 3: Energy levels (2) z x y L1 L2 > L1 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? E (nx,ny,nz) (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. 11Eo 9Eo 6Eo (2,1,1) (1,1,2) D=2 (1,2,1) D=1 3Eo (2) There is an overall lowering of energies due to decreased confinement along y. (1,1,1) D=1