Review of Quantum Physics

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18 CHAPTER 1. REVIEW OF QUANTUM PHYSICS5A. |φ 1 〉 and |φ 2 〉 are normalised eigenfunctions of observable Â. However they are degenerate (havethe same eigenvalue of Â) so are not necessarily orthogonal. If 〈φ 1 |φ 2 〉 = c and c is real, findlinear combinations of φ 1 and φ 2 which are normalised and orthogonal toa) φ 1b) φ 1 +φ 26B. The operators Â1 to Â6 are defined as followsÂ 1 ψ(x) = ψ 2 (x)Â 2 ψ(x) = dψ(x)dxÂ 3 ψ(x) = i dψ(x)dxÂ 4 ψ(x) = x 2 ψ(x)Â 5 ψ(x) = sinψ(x)Â 6 ψ(x) = d2 ψ(x)dx 2where you may assume that ψ(x) is defined over the range −∞ < x < ∞ and that ψ(x) and itsderivative vanish at these limits.Which of these operators are linear? Which are Hermitian?What the eigenfunctions for the linear Hermition operators?7B. A particle in the ground state (n = 1) of a 1D potential well extending from 0 ≤ x ≤ a can bedescribed by the wave function ψ 1 (x) = Asin(πx/a). A particle in the n = 2 state of the sameinfinite potential well has the wavefunction ψ 2 (x) = Bsin(2πx/a)(i) State the normalisation condition satisfied by ψ 1 (x).√2(ii) Show that A =a .(iii) Find B.(iv) What is the probility of finding the particle at x = a/2?(v) Show that ψ 1 (x) = ψ 1 (a−x). What is the name of this type of function?(vi) Show that ψ 2 (x) = −ψ(a−x). What is the name given to this type of function?(vii) Determine the probability that a particle in the n = 1 state resides between x = 0 andx = a/4.(viii) Determine the probability that a particle in the n = 2 state resides between x = 0 andx = a/4(ix) A superposition of the two states is also a possible state i.e. ψ = c 1 ψ 1 +c 2 ψ 2 . Show that|c 1 | 2 +|c 2 | 2 = 1.8B. Determine the normalising constants for the following wavefunctions(i) ψ(x) = A 1 sin(πx/a); 0 ≤ x ≤ a(ii) ψ(x,y,z) = A 2 sin(πx/a)sin(πy/b)sin(πz/c); 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c(iii) ψ(r) = A 3 e −r/a ; for a sphere of infinite radius
1.5. PROBLEMS 199B. A particle is represented by the wavefunctionψ(x) = Axe −αx2Calculate ∆x and ∆p x by making use of the momentum operator. Hence show that ∆x∆p x =3¯h/210B. Show that in one dimension the following wavefunctions are orthogonal ψ 1 = e −x2 , ψ 2 = xe −x2and ψ 3 = (2x 2 −1)e −x2 . How could you make them orthonormal?11B. Which of the following operators are Hermitian, given that Â and ˆB are Hermitian?Â+ ˆB,cÂ,ÂˆB,ÂˆB + ˆBÂShow that in one dimension, for functions that tend to zero at x = ±∞, the operator ∂/∂x isno Hermitian, but the following operator is −i¯h∂/∂x. Is the operator ∂ 2 /∂x 2 Hermitian?12B. The two operators Â and ˆB represent physical observables.(i) Write out the commutation relation [Â, ˆB] in terms of Â and ˆB(ii) If Â and ˆB “commute”, state the relation satisfied by Â and ˆB and explain the physicalsignificance of this result.(iii) What can be said about the eigenfunctions of Â and ˆB if these operators commute?(iv) If Â and ˆB do not commute, what is the physical significance of this result?(v) What can be said about the eigenfunctions of Â and ˆB if these operators do not commute?13C. Observable Â has eigenfunctions ψ 1 and ψ 2 with eigenvalues of a 1 and a 2 . Observable ˆB haseigenfunctions χ 1 and χ 2 with eigenvalues b 1 and b 2 . The eigenfunctions can be related asfollowsχ 1 = (2ψ 1 +3ψ 2 )/ √ 13χ 2 = (3ψ 1 −2ψ 2 )/ √ 13ˆB is measured and a value of b 1 is obtained. If Â were now to be measured, what would be theprobabilities of getting a 1 and a 2 in this measurement? If Â is now indeed measured, and thenˆB measured again, show that the probability of getting b 1 again is 97/16914C. For a certain type of atom the observable Â has eigenvalues ±1, with corresponding eigenfunctionsu + and u − . Another observable ˆB also has eigenvalues of ±1, but correspondingeigenfunctions are ν + and ν − . The eigenfunctions are related byν + = (u + +u − )/ √ 2ν − = (u + −u − )/ √ 2Show that Ĉ ≡ Â+ ˆB is an observable and find the possible results of a measurement of Ĉ.Find the probability of obtaining each result when a measurement of Ĉ is performed on an atomin the state u + , and the corresponding state of the atom immediately after the measurement interms of u + and u − .
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: 1.5. PROBLEMS 17Thus if Â commutes
Page 21: 1.6. ANSWERS TO CHAPTER 1 PROBLEMS

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