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PHYS 221: Homework #04 and Reading Assignment

February 13, 2023

This Written Homework is due Friday, February 17, at 11.59 PM and must be submitted online as a SINGLE CLEARLY LEGIBLE PDF ﬁle via Canvas.

Physics 221: Spring 2023: Week 5 Reading assignment:

• Read P221 Course Documents p221 2023. Book: Modern Physics by Randy Harris - Second Edition, Chapter 5, sections 5. 1 to 5.8 and Chapter 6 sections 6. 1, 6.2,6.4. Review Sheets: Feb 8, Feb 10, Feb 13 and Feb 15, which you can ﬁnd in the Modules section of the PHYS 221 Canvas Page or at the webpage http://ec2-54- 174- 155-225.compute- 1.amazonaws.com:8000

Physics 221: Spring 2023: Homework #04

The Written Homework is due as as scanned PDF ﬁle via Canvas by 11.59 PM, Friday February 17, 2023

Important! Read Course Document #02: Homework Guidelines before you start this homework assignment!

Important note: The Homework will be graded on a scale of 0 to 20 points. Not all problems will be graded. Only a subset of the assigned problems will be graded.

Problem 1. Finite Well

Part (a) Replicate the energy quantization condition (5.22) for the Finite Well considered at section 5.6 Modern Physics by Randy Harris (2nd Ed):

k α

α k

Reproduce equation above by yourself and explain the steps in your own words.

Part (b) From the energy quantization condition (5.22) for the ﬁnite well, recover the energy quantization condition for the inﬁnite-well U0 = &.

Problem 2. The Harmonic Oscillator

The Schroedinger equation for the quantum harmonic oscillator, is given by:

一 + κy2 ψ(y) = Eψ(y) (1)

Part (a) The ﬁrst level state is given by a wave function of the form: ψ 1 (y) = Aye一ay2 /2 .

What should be the value of the parameter a in order for ψ1 be an actual solution of equation (1).

Part (b) By applying the condition of total probability equal one, compute the value of the constant A.

Part (c) The energy level for the harmonic oscillator, are label by an integer n and

as follows:

E = ╱n + 、 ω0 .

Knowing that ψ1 correspond to the ﬁrst level (i.e n = 1), what is ω0 .

Problem 3. Tunneling

During class, we argued that due to the exponential decreasing dependence with respect to the potential height and width, tunneling is very sensitive to the incoming particle energy. Suppose that in a particular wide tunneling process, the exponential approximation for the transmission probability applies:

T ~ 16 ╱ 1 一、e一2L^2m(U0 一E)/ .

Additionally, the height and width of the potential are such that the following

relation is satisﬁed 2L^2mU0 / = 5

Part (a) Calculate the transition probabilities when = 0.4 and when = 0.6.

Part (b) Now do the same but this time for 2L^2mU0 / = 50 and then 2L^2mU0 / =

500

Part (c) How do you results support the claim that the tunneling probability is far more sensitive to the energy E when the tunneling probability is small?

Problem 4. Relativistic dispersion relation

For a relativistic particle, we have that energy and momentum is given respec- tively by E = γumc2 and p = γumu, with u the speed of the particle and m it’s mass. Eliminating u between this two expressions, give us a useful relation between energy and momentum,

E2 = p2 c2 + m2 c4 .

Part (a) From the expression above, compute the quantum dispersion relation for a relativistic particle.

Part (b) Show that in the limit of low speed, the dispersion relation reduces to

mc2 k2

2m .

hint: Write the result in part (a) in terms of the variable α = , which is very small for small speeds (equivalently small k), then used the ﬁrst order Taylor