Review of Quantum Physics

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20 CHAPTER 1. REVIEW OF QUANTUM PHYSICS15B. What is the definition of the commutator [Â, ˆB]?Prove the following commutation relations(i) [ˆp x , ˆp y ] = 0(ii) [ˆp 2 , ˆp x ] = 0(iii) [ˆp x ,Ĥ] = −i¯h∂V∂x(iv) [Ĥ,ˆx] = −i¯h mˆp x16B. In a certain system, Â has eigenvalues a 1 and a 2 corresponding to eigenfunctionsψ 1 = (u 1 +u 2 )/ √ 2 (1.110)ψ 2 = (u 1 −u 2 )/ √ 2, (1.111)where u 1 and u 2 are stationary states with energies E 1 and E 2 . Â is measured and found tohave value a 1 . Find how 〈Â〉 varies with time subsequently17B. (i) State the time-dependent Schrödinger equation.(ii) The time-independent wave function for a 1D system ψ(x) can be expressed in terms of acomplete set of energy eigenfunctions ψ n (x) with corresponding eigenvalues E nψ(x) = ∑ nc n ψ n (x).How can the the time-dependent wavefunction, Ψ(x,t) be expressed in terms of the energyeigenfunctions?(iii) Using your result from part (ii), show that Ψ(x,0), the state of the system at t = 0 isequivalent to the expression for ψ(x) quoted in (ii).
1.6. ANSWERS TO CHAPTER 1 PROBLEMS 211.6 Answers to Chapter 1 Problems1. Oscillating wavefunction where E > V, decaying wavefunction in forbidden regions E < V.2. (i) Linear operators obey Â(af 1(x)+bf 2 (x)) = aÂf 1(x)+bÂf 2(x)(ii) Orthogonal functions obey ∫ ba f 1(x) ∗ f 2 (x)dx = 0 for a particular range (a,b)(iii) Any arbitrary function ψ can be written as a unique sum of a complete set of functionsψ n : ψ = ∑ n c nψ n3. (i) ˆx = x(ii) ˆp x = −i¯h ddx(iii) ˆp y = −i¯h ddy(iv) ˆP = −i¯h∇(v) ˆP 2 = −¯h 2 ∇ 2(vi) ˆT = ˆp2 x2m = − ¯h22md 2dx 2(vii) ˆT = ˆP 22m = − ¯h22m ∇2(viii) ˆV = V(x)(ix) Ĥ = ˆT + ˆV = − ¯h22m dx+V(x) 2d 2(x) Ĥ = ˆT + ˆV = − ¯h22m ∇2 +V(x)4. (i) 〈ψ 1 |cψ 2 〉 = ∫ dxψ ∗ 1cψ 2 = c〈ψ 1 |ψ 2 〉(ii) 〈cψ 1 |ψ 2 〉 = ∫ dx(cψ 1 ) ∗ ψ 2 = c ∗ 〈ψ 1 |ψ 2 〉(iii) 〈ψ 3 |ψ 1 +ψ 2 〉 = ∫ dxψ ∗ 3 (ψ 1 +ψ 2 ) = 〈ψ 3 |ψ 1 〉+〈ψ 3 |ψ 2 〉(iv)ψ = ∑ nc n ψ n5. (a) If ψ = 1 √1−c 2 (−cφ 1 +φ 2 ) then 〈ψ|φ 1 〉 = 0(b) If ψ = 1 √ 2−2c(φ 1 −φ 2 ) then 〈ψ|φ 1 +φ 2 〉 = 0⇒ 〈ψ m |ψ〉 = ∑ c n 〈ψ m |ψ n 〉} {{ }nδ mn⇒ c m = 〈ψ m |ψ〉6. Linear: Â 2 ,Â3,Â4,Â6Hermitian: Â 3 ,Â4,Â6Eigenfunctions (Âψ = λψ):Â 3 : ψ(x) = e −iλxÂ 4 : ψ(x) = δ(x−x 0 )Â 4 : ψ(x) = Ae √ λx +Be −√ λx
Page 5 and 6: 1.2. EXAMPLE: INFINITE SQUARE WELL
Page 7 and 8: 1.2. EXAMPLE: INFINITE SQUARE WELL
Page 9 and 10: 1.3. DIRAC NOTATION 9tP(x,t)=| ψ |
Page 11 and 12: 1.4. THE POSTULATES 11Classical Var
Page 13 and 14: 1.4. THE POSTULATES 13so here A n a
Page 15 and 16: 1.4. THE POSTULATES 15Labelling qua
Page 17 and 18: 1.5. PROBLEMS 17Thus if Â commutes
Page 19: 1.5. PROBLEMS 199B. A particle is r

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