Quantum Mechanics Â¯h = 1.0545716 10 kg m sec

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Contents 0 Words 5 0.1 Quantum mechanics past, leading to present . . . . . . . . . . . . 5 0.2 A road map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.3 Exercise 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 States 9 1.1 The state of a system : ket notation . . . . . . . . . . . . . . . . 9 1.2 The superposition principle . . . . . . . . . . . . . . . . . . . . . 10 1.3 Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 The probability interpretation . . . . . . . . . . . . . . . . . . . . 13 1.6 The measurement problem . . . . . . . . . . . . . . . . . . . . . . 14 1.7 Bra’s and inner products : the Hilbert space . . . . . . . . . . . . 15 1.8 Playing with light . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.9 Exercises 1 to 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Observables 23 2.1 A measurement with a specific result . . . . . . . . . . . . . . . 23 2.2 A measurement with a general result . . . . . . . . . . . . . . . . 24 2.3 Using probability theory . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Hermitean operators and eigenstuff . . . . . . . . . . . . . . . . . 26 2.5 Observables as matrices . . . . . . . . . . . . . . . . . . . . . . . 28 2.6 This is not a measurement ! . . . . . . . . . . . . . . . . . . . . . 29 2.7 Degeneracy of eigenvalues . . . . . . . . . . . . . . . . . . . . . . 30 2.8 What is a classical observable ? . . . . . . . . . . . . . . . . . . . 30 2.9 Exercises 7 to 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Systems of observables 36 3.1 More than one property . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 The commutator of two operators . . . . . . . . . . . . . . . . . . 36 3.3 Commensurable observables and common bases . . . . . . . . . . 37 3.4 Complete commensurable sets of observables . . . . . . . . . . . 38 3.5 Incommensurability and information loss . . . . . . . . . . . . . . 39 3.6 Commensurability and state labelling . . . . . . . . . . . . . . . 40 3.7 The Heisenberg inequality . . . . . . . . . . . . . . . . . . . . . . 41 3.8 Extra ! The Mermin-Peres magic square . . . . . . . . . . . . . 43 3.9 Exercises 17 to 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 Does reality have properties ? 47 4.1 The question at issue . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 The EPR paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2.1 EPR’s thought experiment . . . . . . . . . . . . . . . . . 47 4.2.2 Entangled states . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Bell’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2
4.3.1 The experiment : Obama vs Osama . . . . . . . . . . . . 49 4.3.2 Hidden variables : Bell’s inequality . . . . . . . . . . . . . 49 4.3.3 QM’s predictions . . . . . . . . . . . . . . . . . . . . . . . 51 4.3.4 Faster than light ? . . . . . . . . . . . . . . . . . . . . . . 52 4.3.5 Extra ! A loophole ? The no-clone theorem . . . . . . . 53 5 The Schrödinger equation 55 5.1 The Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Preservation of the norm . . . . . . . . . . . . . . . . . . . . . . 56 5.3 The time-independent Schrödinger equation . . . . . . . . . . . . 56 5.4 Time evolution : the Heisenberg equation . . . . . . . . . . . . . 58 5.5 The free particle and the fundamental commutator . . . . . . . . 58 5.6 The Ehrenfest theorem . . . . . . . . . . . . . . . . . . . . . . . . 60 5.7 Extra ! Maximum speed : the Margolus-Levitin theorem . . . 61 5.8 Exercises 22 to 29 . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6 The wave function 66 6.1 The position representation . . . . . . . . . . . . . . . . . . . . . 66 6.2 Position and momentum for wave functions . . . . . . . . . . . . 67 6.3 Inner products for wave functions . . . . . . . . . . . . . . . . . . 68 6.4 The Schrödinger equation for wave functions . . . . . . . . . . . 69 6.5 Doing the wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.6 Extra ! Probability currents . . . . . . . . . . . . . . . . . . . 71 6.7 Particle in a box : energy quantisation . . . . . . . . . . . . . . . 73 6.8 Extra ! Antimatter . . . . . . . . . . . . . . . . . . . . . . . . 75 6.9 Exercises 30 to 45 . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7 The Harmonic oscillator 84 7.1 A general phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 84 7.2 Raising and lowering operators . . . . . . . . . . . . . . . . . . . 85 7.3 Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.4 A Hamiltonian and its spectrum . . . . . . . . . . . . . . . . . . 87 7.5 The Hamiltonian in operator form . . . . . . . . . . . . . . . . . 87 7.6 Wave functions for the harmonic oscillator . . . . . . . . . . . . . 88 7.7 Expectation values of position and momentum . . . . . . . . . . 90 7.8 Extra ! Coherent states . . . . . . . . . . . . . . . . . . . . . . 91 7.9 Exercises 46 to 52 . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8 Other one-dimensional systems 96 8.1 A kinky function . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.2 Pinning down a particle : the single pin . . . . . . . . . . . . . . 97 8.3 The double-pin model . . . . . . . . . . . . . . . . . . . . . . . . 98 8.4 A single barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.5 The double barrier : a wave filter . . . . . . . . . . . . . . . . . . 103 8.6 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 8.7 Extra ! Periodic potentials . . . . . . . . . . . . . . . . . . . . 106 3
Page 1: Quantum Mechanics Ronald Kleiss (R.
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
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leak, however. In the first place,
Page 55 and 56:
5 The Schrödinger equation — I h
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Suppose that the spectrum of E cons
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in one dimension without any forces
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where we have used the fundamental
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5.8 Exercises 22 to 29 Exercise 22
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Exercise 28 : Touch, don’t cut Ve
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information about the system as doe
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conjugate, while for the ket we tak
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where p is the value of the momentu
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6.7 Particle in a box : energy quan
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to dE dL = −2E L . (178) That is,
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x A E < 0 B t A particle with negat
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antiparticle have annihilated. Howe
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1. Perform the p integral and show
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2. Show that and that φ cl = arcta
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We shall now discuss the quantum me
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7.4 A Hamiltonian and its spectrum
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Note that we have now constructed a
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and we can check Heisenberg’s ine
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simplicity. Suppose that we want to
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(a) Suppose that the state is |s〉
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8.2 Pinning down a particle : the s
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10 8 6 4 2 Here we plot, for r = 0.
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thus have the following forms for t
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8.5 The double barrier : a wave fil
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while inside the barrier − ¯h2 2
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A general eigenfunction with energy
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Unsurprisingly, a drops out since i
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put everything together. In the plo
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Exercise 60 : A potential step We c
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9.3 Global and internal observables
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These two equations, both of the fo
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3. Show that the system is still se
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Exercise 69 : Stuck with one anothe
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= 〈 a 2 − 2a 〈a〉 + 〈a〉
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12 The Gaussian integral Let us con
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where we have used z = t 2 and the
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so that we find that in all cases F
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3. 〈ν µ |ν e ; t〉 = ( ) ( )
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1. By its definition, 〈k〉 = 6
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and each of them has eigenvalues 1
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then i¯h d dt |ψ C〉 = Ce −iCt
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Solution to Exercise 28 Eq.(137) ca
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with A = − (p − p 0) 2 4σ 2
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if we drop an overall complex phase
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14.6 Excercises 46 to 52 Solution t
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= 1 ( ) s 2 + (s ∗ ) 2 + 2s ∗ s
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Also, we have ψ(0) = 1 − e −α
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equations themselves, and arrive at
show all

Quantum Mechanics Â¯h = 1.0545716 10 kg m sec

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