An electron's state is the (improper

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Position, momentum and the origin of the canonical commutator

Problem

An electron's state is the (improper, delta-normalised) momentum eigenstate

\varphi_{p_0}(x) = \langle x|p_0\rangle = \dfrac{1}{\sqrt{2\pi\hbar}}\,e^{ip_0x/\hbar},

with

p_0 = h/\lambda

for a de Broglie wavelength

\lambda = 0.40

nm. ((a)) Using

\langle x|\hat T(a)|\varphi_{p_0}\rangle = \varphi_{p_0}(x-a)

, show that

|p_0\rangle

is an eigenvector of

\hat T(a)

, and find its eigenvalue for a translation

a=0.10

nm. ((b)) Give the phase angle of this eigenvalue in radians, and confirm that its modulus is

1

. ((c)) Explain in one sentence what this modulus tells about the probability density

|\varphi_{p_0}(x)|^2

under translation.