Chapter 5 wrote a state of the infinite well as with

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Free particles, wave packets and tunnelling

Problem

Chapter 5 wrote a state of the infinite well as

\Psi(x,0) = \sum_n c_n\psi_n(x)

with

c_n = \int_0^a\psi_n(x)\Psi(x,0)\,dx

. This chapter writes a free particle as

\Psi(x,0) = (2\pi)^{-1/2}\int\phi(k)e^{ikx}dk

with

\phi(k) = (2\pi)^{-1/2}\int\Psi(x,0)e^{-ikx}dx

. ((a)) Match each piece of the well's expansion to its counterpart in the packet's: what plays the role of

\psi_n

? Of

c_n

? Of the sum over

n

? ((b)) A packet is prepared with

\Psi(x,0) = (2\pi s^2)^{-1/4}e^{-x^2/4s^2}e^{ik_0x}

s = 0.20

nm and

k_0 = 15

^{-1}

. Using the Gaussian result from this chapter, evaluate

\phi(k)

k = k_0

and at

k = k_0 + 1/2s

, and say what feature of

\phi

these two values mark out.