Review of Quantum Physics

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 follows 1 ψ(x) = ψ 2 (x) 2 ψ(x) = dψ(x)dx 3 ψ(x) = i dψ(x)dx 4 ψ(x) = x 2 ψ(x) 5 ψ(x) = sinψ(x) 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