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 1. What do we solve for? Energy levels 2. 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 1. Potential Energy will depend on position, not time 2. We will deal with relativity later. 3. We will ensure that the wavefunction, , and its first derivative with respect to position are continuous.   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,  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  (-x)=  (x)

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

13. Writing a general function for k

14. Wavefunctions for Even and Odd

15. Finding A n and B n

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. Finite Square The finite square well can be made more realistic by assuming the walls of the container are a finite height, V o. Region 1Region 3 x= -a x=ax V(x) V0V0 Region 2

25. The Schroedinger Equations

26. The Boundary Conditions  1 (-a)=  2 (-a)  2 (a)=  3 (a)  1 ’(-a)=  2 ’(-a)  2 ’(a)=  3 ’(a) 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 these 2 equations could be solved simultaneously for  and , 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 V o is assumed. The number of excited bound states increases with V o (the radius of the circle) and the states of opposite parity are interleaved. If V o becomes very large, the values of  approach n  /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>> 

38. Constructing a trial solution

39. Adding terms

40. Sol’ns of H are a power series

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

42. How to generate a j ’s For odd parity i.e. q 1,q 3,q 5, start with a o =0 and a 1 <>0 For even parity i.e. q 0,q 2,q 4, start with a o <>0 and a 1 =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(-q 2 /2) So we must truncate the power series (make it finite) The easiest way is to make a j+2 vanish This occurs when  -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)  , 1/2   for each degree of freedom

45. Summary so far: The solutions, H n, 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 H n is 2 n E.g. For H 5, the coefficient for q 5 is 2 5 Also, recall for n=5, E=(5+1/2)   or 11/2   At this eigenvalue, H= a 1 q+a 3 q 3 +a 5 q 5 where a 5 =32

47. Method 1: Recursion Formula The current convention is that the coefficient for the n- th degree term of H n is 2 n E.g. For H 5, the coefficient for q 5 is 2 5 Also, recall for n=5, E=(5+1/2)   or 11/2   At this eigenvalue, H= a 1 q+a 3 q 3 +a 5 q 5 where a 5 =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 H n+1 =2*q*H n -2*n*H n-1 We know that H 0 =1 and H 1 =2q Ideally suited for computers Also, the derivative with respect to q is H’ n =2*n*H n-1

51. Method 5: Table Look Up

52. Other Methods Continued Fractions Schmidt Orthogonalization

53. Integrals with Hermite Polynomials

54. We know that the biggest term of H n is 2 n q n H n ~2 n q n H’ n ~2 n *n*q n-1 H’’ n ~2 n *n*n-1*q n-2 H n- ’=2 n *n!*q o =2 n *n!

55. Going back and using the n-th derivative

56. Now, we have everything

57. Transition Matrix Elements

58. SHO via Operators (or via Algebra) While the analytical solution is one way to solve this problem, using operators 1. Teaches us the operator method which will be required when we study the quantum theory of angular momentum as well as quantum field theory 2. Gives us another tool in our toolbox for problem solving.

59. Many Slides ago, Which is good Schroedinger notation but what about Dirac?

60. In Dirac notation, Where |n> is the n th state of the oscillator

61. Even More Definitions,

62. A new way to TISE This is TISE in operator form

63. A necessary aside

64. Gosh, more definitions?

65. It turns out that

66. Let’s solve for q 2 +p 2

67. A crazy form of TISE

68. Theorem 8 a + is called the “raising operator”

69. Proof of Thm 8

70. Theorem 9 a is called the “lowering operator”

71. Proof of Thm 9 See “Proof of Thm 8” and put minus signs in the appropriate places

72. Theorem 10   1

73. Consequences We could start with state |n> and lower it until we reached a ground state i.e. a(a|n>)=(  -4)|n> etc.

74. Theorem 11 In the ground state,  =1

75. Now, let’s find the wavefunction of the ground state (hint: it must agree with what we found earlier H o exp(-q 2 /2) A 0 is found by normalizing the wavefunction and this is exactly in agreement with the analytical method.

76. Now other states, use a + on |0> and raise, raise, raise A n is found by normalization

77. Calculating Transition Matrix Elements

78. So the matrices look like

79. The matrices of q and p are

80. What about matrices of q 2 and p 2 ?

81. The Dirac Delta The Dirac delta,  (x), is used to represent a point particle  (x)=0 if x<>0  (x)=  if x=0 But what keeps it finite, is the normalization

82. The Dirac Delta cont’d If it is at x=a  (x-a)=0 if x<>a  (x-a)=  if x=a

83. TISE for the Dirac Delta

84. Some Boundary Conditions At x=|  |,  =0 The big problem lies at x=0 It turns out that  is continuous But  ’ is discontinuous X=0 Technically, this is infinitesimally thick. So the derivatives change from + to – in an infinitesimal distance

85. We could integrate over a small space X=0 X=+  X=-  So the idea is to integrate from –  to  and then let  go to zero

86. Mathematically

87. Using an ansatz

88. Plugging  into the TISE

89. Solving for E Note that energy is a single value; based on the potential amplitude, A

90. Find 

91. The Uniform Force Field V(x)=G*x, x>0 V(x)=0, x<0 G is a positive constant equal to the gradient of the potential Function has several physical examples: An electric charge in an uniform field near an impermeable plate A tennis ball dropped down an elevator shaft (hence the name of the quantum bouncer)

92. TISE

93. A change of variables

94. Airy’s Functions

95. More about Airy

96. The root of the matter The “roots” of a function are the values where a function is equal to zero We solve the previous equation for E We set z o equal to the roots of the Airy function, n nRoot 12.3381 24.08794 35.52055 46.7867 57.94413 69.02265 710.04017 811.00852 911.93601 1012.82877

97. Graphically,

98. General Features of 1-D bound states   vanishes at |x|=  2. 1-d Bound states are non-degenerate By degenerate, 2 states have equal energy 3. Wave functions for a 1-d bound state can be constructed so that it is real 4. For a 1-d bound state, =0 5. If H is symmetric, the wave functions are eigenfunctions of the parity operator 6. The Schroedinger Equation can be converted into an integral equation.

99. Proof of #2: Nondegeneracy of 1-d bound states

100. Proof of #3: Wavefunctions can be constructed to be real

101. Proof of #4: =0

102. Proof of #5: Wavefunctions are eigenfunctions of parity operator

103. Expansion on #6: