Mr. A. Square’s Quantum Mechanics

1. Mr. A. Square’s Quantum Mechanics Part 1

2. Bound States in 1 Dimension

3. The Game’s Afoot • Primarily, the rest of this class is to solve the time independent Schroedinger Equation (TISE) for various potentials • What do we solve for? • Energy levels • Why? • Because we can measure energy most easily by studying emissions and absorptions

4. 2) What type of potentials? • We will start slow– with 1-dimensional potentials which have limited physicality i.e. not very realistic • We then take a detour into the Quantum Theory of Angular Momentum • Using these results, we then study 3-dimensional motion and work our way up to the hydrogen atom….

5. Some of these potentials seem a little … dumb. • While a particular potential may not seem realistic, they teach how to solve the harder, more realistic problems. We learn a number of tips and tricks.

6. The Big Rules • Potential Energy will depend on position, not time • We will deal with relativity later. • We will ensure that the wavefunction, y, and its first derivative with respect to position are continuous. • y will have reasonable values at the extremes

7. Mathematically,

8. Re-writing Schroedinger • A more easily form of the Schroedinger equation is as follows:

9. Infinite Square Well

10. Apply Boundary conditions • At x=-a, y must be zero in order to satisfy our continuity condition; also, at x=a.

11. Even Solutions • Assume A <>0, then B is zero • These are called “even” functions since • y(-x)=y(x)

12. Odd Solutions • Assume B <>0, then A is zero • These are called “odd” functions since • y(-x)=-y(x) • Evenness or oddness has to do with “parity”

13. Writing a general function for k

14. Wavefunctions for Even and Odd

15. Finding Anand Bn

16. Wavefunctions for Even and Odd

17. Wavefunctions

18. In the momentum representation

19. Matrix Elements of the Infinite Square Well • The matrix elements of position, x, are of interest since they will determine the dipole selection rules. • The matrix element is defined as

20. Several Transitions

21. If I write the wavefunctions as

22. Then I can write x as a m x n matrix

23. The momentum operator is derived in a similar manner but is imaginary and anti-symmetric

24. V(x) V0 Region 1 Region 3 x= -a x=a x Finite Square • The finite square well can be made more realistic by assuming the walls of the container are a finite height, Vo. Region 2

25. The Schroedinger Equations

26. y1(-a)=y2(-a) y2(a)=y3(a) y1’(-a)=y2’(-a) y2’(a)=y3’(a) The Boundary Conditions • The only way to satisfy all four equations is to have either B or C vanish

27. If C=0, then we have states of even parity

28. If C=0, then we have states of even parity • If these 2 equations could be solved simultaneously for k and K, then E could be found. • Two options: • Numerically (a computer) • Graphically

29. Even Parity Solutions Each intersection represents a solution to the Schroedinger equation k=6.8 K= 4.2

30. Odd Parity Solutions Each intersection represents a solution to the Schroedinger equation

31. Inspecting these graphs • Note that the ground state always has even parity (i.e. intersection at 0,0), no matter what value of Vo is assumed. • The number of excited bound states increases with Vo (the radius of the circle) and the states of opposite parity are interleaved. If Vo becomes very large, the values of k approach np/2a • This asymptotic approach agrees which the quantization of energy in the infinite square well

32. The Harmonic Oscillator • It is useful in describing the vibrations of atoms that are bound in molecules; in nuclear physics, the 3-d version is the starting point of the nuclear shell model • We will solve this problem in two different ways: • Analytical (integrating as we have done before) • Using Operators • But first we need some definitions

33. Angular frequency

34. TISE for SHO

35. More Definitions

36. Change of Variables

37. Let q>>e

38. Constructing a trial solution

39. Adding terms

40. Sol’ns of H are a power series

41. Finding aj’s Note: These equation connects terms of the same parity i.e 1,3,5 or 0,2,4

42. How to generate aj’s • For odd parity i.e. q1,q3,q5, start with ao=0 and a1<>0 • For even parity i.e. q0,q2,q4, start with ao<>0 and a1=0 • We will learn an established procedure in a few pages

43. Too POWERful! • We have a problem, an infinite power series has an infinite value • So we know that the series at large values must approximate exp(-q2/2) • So we must truncate the power series (make it finite) • The easiest way is to make aj+2 vanish • This occurs when e-1-2j=0

44. Making j equal to n • This is a big result, since you know from Modern Physics that we had to get this result. • For a 3-d SHO, E=(n+3/2)w, 1/2 w for each degree of freedom

45. Summary so far: • The solutions, Hn, to the differential equation are called “Hermite” polynomials. • In the next few slides, we will learn how to generate them

46. Method 1: Recursion Formula • The current convention is that the coefficient for the n-th degree term of Hn is 2n • E.g. For H5, the coefficient for q5 is 25 • Also, recall for n=5, E=(5+1/2)w or 11/2 w • At this eigenvalue, H= a1q+a3q3+a5q5where a5=32

47. Method 1: Recursion Formula • The current convention is that the coefficient for the n-th degree term of Hn is 2n • E.g. For H5, the coefficient for q5 is 25 • Also, recall for n=5, E=(5+1/2)w or 11/2 w • At this eigenvalue, H= a1q+a3q3+a5q5where a5=32

48. Method 2: Formula of Rodriques

49. Method 3: Generating Function • The generating function is a function of two variables, q and s, where s is an auxiliary function • To use this function, take the derivative n times and set s equal to 0

50. Method 4: Recurrance Relations • Hn+1=2*q*Hn-2*n*Hn-1 • We know that H0=1 and H1=2q • Ideally suited for computers • Also, the derivative with respect to q is • H’n =2*n*Hn-1