Quantum Mechanics Â¯h = 1.0545716 10 kg m sec

More documents

Recommendations

Info

2. Show that [ˆ⃗ L, ˆ⃗x 2 ] = [ˆ⃗L, ˆ⃗p 2 ] = 0 Do this componentwise, for instance by computing [ˆL x , ˆx 2 + ŷ 2 + ẑ 2 ]. 3. Show that, for a central potential, [ˆ⃗ L, Ĥ] = 0 and that therefore also angular momentum is conserved. 4. Show that the components of angular momentum are incommensurable (this is easy!). Show that, on the other hand, [ˆ⃗ L, ˆ⃗L 2 ] = 0 5. Show the following : if a particle moves under the influence of a central potential, and we have measured the values of ⃗ L 2 and L z , say, these values will be preserved throughout the motion of the particle. 6. Show that for such a particle a maximal commensurable set of observables is formed by E, ⃗ L 2 and ⃗ L. Exercise 67 : Three-dimensional harmonic atom Generalize the model of Exercises 62 and 63 to the case of three dimensions.. Be careful ! Both the result for the total energy and that of the degeneracy changes. Exercise 68 : Degeneracies of non-isotropic harmonic oscillators Consider a three-dimensional harmonic oscillator that is non-isotropic, which means that the oscillation frequencies are not the same in all directions. The Hamiltonian is Ĥ = 2mˆ⃗p 1 2 + m ( ωxˆx 2 + ω y ŷ 2 + ω z ẑ 2) 2 1. Show that this system is separable, and give the form of the energy eigenvalues. 2. Suppose that the frequencies are in the proportion ω x : ω y : ω z = 1 : √ 2 : √ 3 Show that there is no degeneracy. (Hint : √ 6 is irrational). 3. Let the frequencies be in the proportion ω x : ω y : ω z = 1 : 1 : 2 Compute the degeneracies of the five lowest energies. general formula. Try to find the 120
Exercise 69 : Stuck with one another We consider a one-dimensional system of two particles. The particles can move freely except that they can not be further apart than a distance L (imagine the particles tied together by a chain of length L). 1. Find the potential that describes this situation, and show that it is translationinvariant. 2. Let the particles have mass m and 2m respectively. Give the Hamiltonian for this system in terms of the particles’ position and momentum. 3. Go over to global and internal position and momenta, and give the Hamiltonian in terms of these. 4. Derive the form of the total energy of the system. 121
Page 1 and 2:
Quantum Mechanics Ronald Kleiss (R.
Page 3 and 4:
4.3.1 The experiment : Obama vs Osa
Page 5 and 6:
0 Words — All science is either p
Page 7 and 8:
mechanics their rôle is taken over
Page 9 and 10:
1 States — Can nature possibly be
Page 11 and 12:
If |ψ 1 〉 and |ψ 2 〉 are two
Page 13 and 14:
of states is called a basis. In our
Page 15 and 16:
tried to either explain, evade, or
Page 17 and 18:
show precisely the properties of st
Page 19 and 20:
since (apparently) the states |0〉
Page 21 and 22:
Exercise 4 : Some inner products Co
Page 23 and 24:
2 Observables — To observe is to
Page 25 and 26:
not just the position coordinate X
Page 27 and 28:
(whatever all its ingredients may m
Page 29 and 30:
is nothing but the Hermitean conjug
Page 31 and 32:
The distribution of measurement val
Page 33 and 34:
in Eq.(34) the sum is correct : ∑
Page 35 and 36:
Exercise 15 : Eigenvalues and degen
Page 37 and 38:
But this implies that Â ˆB |a n ,
Page 39 and 40:
It is one of the peculiar features
Page 41 and 42:
we choose A, a complete description
Page 43 and 44:
3.8 Extra ! The Mermin-Peres magic
Page 45 and 46:
1. Prove the identities given in Eq
Page 47 and 48:
4 Does reality have properties ? 4.
Page 49 and 50:
4.3 Bell’s inequalities In 1964,
Page 51 and 52:
never decrease : ∫ C(θ 1 , θ 2
Page 53 and 54:
leak, however. In the first place,
Page 55 and 56:
5 The Schrödinger equation — I h
Page 57 and 58:
Suppose that the spectrum of E cons
Page 59 and 60:
in one dimension without any forces
Page 61 and 62:
where we have used the fundamental
Page 63 and 64:
5.8 Exercises 22 to 29 Exercise 22
Page 65 and 66:
Exercise 28 : Touch, don’t cut Ve
Page 67 and 68:
information about the system as doe
Page 69 and 70: conjugate, while for the ket we tak
Page 71 and 72: where p is the value of the momentu
Page 73 and 74: 6.7 Particle in a box : energy quan
Page 75 and 76: to dE dL = −2E L . (178) That is,
Page 77 and 78: x A E < 0 B t A particle with negat
Page 79 and 80: antiparticle have annihilated. Howe
Page 81 and 82: 1. Perform the p integral and show
Page 83 and 84: 2. Show that and that φ cl = arcta
Page 85 and 86: We shall now discuss the quantum me
Page 87 and 88: 7.4 A Hamiltonian and its spectrum
Page 89 and 90: Note that we have now constructed a
Page 91 and 92: and we can check Heisenberg’s ine
Page 93 and 94: simplicity. Suppose that we want to
Page 95 and 96: (a) Suppose that the state is |s〉
Page 97 and 98: 8.2 Pinning down a particle : the s
Page 99 and 100: 10 8 6 4 2 Here we plot, for r = 0.
Page 101 and 102: thus have the following forms for t
Page 103 and 104: 8.5 The double barrier : a wave fil
Page 105 and 106: while inside the barrier − ¯h2 2
Page 107 and 108: A general eigenfunction with energy
Page 109 and 110: Unsurprisingly, a drops out since i
Page 111 and 112: put everything together. In the plo
Page 113 and 114: Exercise 60 : A potential step We c
Page 115 and 116: 9.3 Global and internal observables
Page 117 and 118: These two equations, both of the fo
Page 119: 3. Show that the system is still se
Page 123 and 124: = 〈 a 2 − 2a 〈a〉 + 〈a〉
Page 125 and 126: 12 The Gaussian integral Let us con
Page 127 and 128: where we have used z = t 2 and the
Page 129 and 130: so that we find that in all cases F
Page 131 and 132: 3. 〈ν µ |ν e ; t〉 = ( ) ( )
Page 133 and 134: 1. By its definition, 〈k〉 = 6
Page 135 and 136: and each of them has eigenvalues 1
Page 137 and 138: then i¯h d dt |ψ C〉 = Ce −iCt
Page 139 and 140: Solution to Exercise 28 Eq.(137) ca
Page 141 and 142: with A = − (p − p 0) 2 4σ 2
Page 143 and 144: if we drop an overall complex phase
Page 145 and 146: 14.6 Excercises 46 to 52 Solution t
Page 147 and 148: = 1 ( ) s 2 + (s ∗ ) 2 + 2s ∗ s
Page 149 and 150: Also, we have ψ(0) = 1 − e −α
Page 151 and 152: equations themselves, and arrive at
show all

Quantum Mechanics Â¯h = 1.0545716 10 kg m sec

Create successful ePaper yourself

Delete template?

Save as template?