Schrodingers Equation for Three Dimensions. QM in Three Dimensions The one dimensional case was good for illustrating basic features such as quantization. - презентация

1 Schrodingers Equation for Three Dimensions

2 QM in Three Dimensions The one dimensional case was good for illustrating basic features such as quantization of energy.

3 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.

4 Schrödinger's Equa 3Dimensions For 3-dimensions Schrödinger's equation becomes,

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

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

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

8 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,

9 Particle in a 3 Dimensional Box

10 The simplest case is a particle confined to a cube of edge length L.

11 Particle in a 3 Dimensional Box

12 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.

13 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.

14 Particle in a 3 Dimensional Box Note: If we consider one coordinate the solution will be the same as the 1-D box.

15 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):

16 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,

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

18 Particle in a 3 Dimensional Box Clearly

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

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

21 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

22 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,

23 Particle in a 3 Dimensional Box with and so forth.

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

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

26 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.

27 Particle in a 3 Dimensional Box Note that three quantum numbers are required to describe the quantum state of the system.

28 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.

29 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.

30 Particle in a 3 Dimensional Box The total energy will be quoted in the form

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

32 Particle in a 3 Dimensional Box Degeneracy

33 Particle in a 3 Dimensional Box Degeneracy: quantum levels (different quantum numbers) having the same energy.

34 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).

35 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.

36 Particle in a 3 Dimensional Box There are three 1 st excited states having the same energy. They correspond to combinations of the quantum numbers whose squares sum to 6.

37 Particle in a 3 Dimensional Box There are three 1 st excited states having the same energy. They correspond to combinations of the quantum numbers whose squares sum to 6. That is

38 Particle in a 3 Dimensional Box The 1 st five energy levels for a cubic box. n2n2 Degeneracy 12none none 4E 0 11/3E 0 2E 0 3E 0 E0E0

39 Schrödinger's Equa 3Dimensions The formulation in cartesian coordinates is a natural generalization from one to higher dimensions.

40 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.

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

42 Consider a particle in a two-dimensional (infinite) well, with L x = L y. 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., L x < L y ) 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

43 Consider a particle in a two-dimensional (infinite) well, with L x = L y. 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., L x < L y ) 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) = E 0 ( ) = 10 E 0 E (2,2) = E 0 ( ) = 8 E 0

44 Example 2: Energy levels (1) Now back to a 3D cubic box: Show energies and label (n x,n y,n z ) for the first 11 states of the particle in the 3D box, and write the degeneracy D for each allowed energy. Use E o = h 2 /8mL 2. z x y L L L E 44

45 z x y L L L D=1 6E o (2,1,1) (1,2,1) (1,1,2) D=3 E (1,1,1) 3E o (n x,n y,n z ) n x,n y,n z = 1,2,3,... Example 2: Energy levels (1) Now back to a 3D cubic box: Show energies and label (n x,n y,n z ) for the first 11 states of the particle in the 3D box, and write the degeneracy D for each allowed energy. Use E o = h 2 /8mL 2.

46 E 3E o 6E o 9E o 11E o (n x,n y,n z ) z x y L1L1 L 2 > L 1 L1L1 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 cubes energy levels below?

47 (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=1 D=2 (2,1,1) (1,1,2) (2) There is an overall lowering of energies due to decreased confinement along y. E 3E o 6E o 9E o 11E o (n x,n y,n z ) 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 cubes energy levels below? z x y L1L1 L 2 > L 1 L1L1