e Physics homework help. Physics 4C — Final Exam

Note: Instructions and rules are as listed on Canvas.

You may mark your solutions directly in the spaces provided below or write them on separate

sheets of paper. In any case, all solutions must be uploaded before the stated deadline.

Name:

Reference Materials

The constants, equations, and mathematical notes below are for your reference during the exam.

Well-prepared students will likely use this infrequently, and then only to find the exact form of a

relationship they are already aware of and planning to use. Do not take the presence or absence of

any specific thing here to be of any consequence, though if you feel it is missing something absolutely

critical to your success on the exam you may ask the instructor.

c = 3 × 108 m/s; me = 9.11 × 10−31 kg = 0.511 MeV/c

2

; e = 1.6 × 10−19 C

h = 6.626 × 10−34 J · s = 4.136 × 10−15 eV · s; ~ =

h

2π

e

iθ = cos θ + isin θ; sin(α ± β) = sin α cos β ± sin β cos α; cos(α ± β) = cos α cos β ∓ sin α sin β

sin2

θ =

1

2

(1 − cos(2θ)) ; cos2

θ =

1

2

(1 + cos(2θ))

vn =

c

n

; n1 sin θ1 = n2 sin θ2; I = I0 cos2

θ

f =

R

2

;

1

do

+

1

di

=

1

f

; m =

hi

ho

= −

di

do

;

1

f

=

n2

n1

− 1

1

R1

−

1

R2

∆` = mλ or

m +

1

2

λ; ∆` = d sin θ; ym =

mλD

d

a sin θ = nλ; θ = 1.22

λ

D

β =

v

c

; γ =

1

p

1 − β

2

; ∆t = γτ ; L =

L0

γ

; ct0 = γ(ct − βx); x

0 = γ(x − βct); y

0 = y; z

0 = z

ux =

u

x + v

1 + u

x

v/c2

; uy =

u

y

/γ

1 + u

x

v/c2

; uz =

u

z

/γ

1 + u

x

v/c2

~p = γm~u; E = γmc2

; KE = (γ − 1)mc2

; E

2 = (pc)

2 + (m0c

2

)

2

λmaxT = 2.898 × 10−3 m · K; Eγ = hf =

hc

λ

; p =

h

λ

; p = ~k; KEmax = hf − φ

∆λ =

h

mec

(1 − cos θ); En = −

13.6 eV

n2

; ∆x∆p ≥

~

2

P(−∞ < x < ∞) = 1; P(a < x < b) = Z b

a

|Ψ(x, t)|

2

dx

−

~

2

2m

∂

2Ψ(x, t)

∂x2

+ U(x, t)Ψ(x, t) = i~

∂Ψ(x, t)

∂t ; −

~

2

2m

d

2ψ(x)

dx2

+ U(x)ψ(x) = Eψ(x)

Note: Short-form questions are viewed and submitted directly in the MyOpenMath assessment.

Physics 4C Final Exam, Page 1 of 8 Spring 2020

Free-Response Questions

Carefully work through the following three free-response questions, each of which has four parts. Do

not leave anything blank! At a minimum, try to write down the given quantities and anything

else that you think is relevant. Each part is worth 5 points, for a total of 60 points in this section.

1. Optics: Please complete the following series of questions on optics.

(a) A converging lens (focal length: f) and an object at a distance x are set up along an axis

so that a real image forms (x > f) on the other side of the lens. Derive the distance (from

the lens) and magnification of the image, both as a function of the object distance x. Is

the image upright or inverted?

(b) Suppose that the object from part (a) is moving toward the lens at a constant speed v.

Find an expression for the speed of the image as a function of the object’s distance x

(measured from the lens, as a positive quantity). [Hint: you may look up the quotient

rule, if you need to.]

Physics 4C Final Exam, Page 2 of 8 Spring 2020

(c) In studying optics, we often made use of the equations d sin θ = mλ and a sin θ = nλ.

Compare and contrast these equations. Be sure to (i) define the meaning of each variable

that appears (a diagram may be helpful), (ii) indicate in which situation (i.e., diffraction

or interference) each should be used, and (iii) describe the nature of light intensity (i.e.,

maxima or minima) that each is relevant for. Using your answers, describe what we mean

when we say “missing order” in the context of diffraction and interference.

(d) On the first exam of the semester, you derived an expression for the intensity of the pattern

on a distant screen that results from interference after light of wavelength λ is passed

through two slits, separated by a distance d. In doing so, we inherently assumed that the

two slits had the same size (and that the size of each was such that diffraction could be

ignored). Here, we’ll maintain the assumption that diffraction can be ignored, but we’ll

remove the assumption that the slits have the same size (and thus, remove the assumption

that the intensity coming through each slit is the same). How would the interference

pattern change if one of the slits is 50% larger than the other? It is not necessary to do

any rigorous derivations here, but make sure to explain how the intensity maxima and

minima will noticeably change (or not change) relative to the equal-width case. Justify

any statements you make using sound physical reasoning.

Physics 4C Final Exam, Page 3 of 8 Spring 2020

2. Special Relativity: In the following series of questions, we’ll consider a ladder of proper length

Lladder = 10 m traveling with speed v relative to the ground (the ladder is oriented parallel to

the ground as well). At rest relative to the ground is a barn of length Lbarn = 5 m.

(a) Show that, from the perspective of an observer riding with the ladder, it will never fit

inside the barn no matter what its speed is.

(b) Now show that, from the perspective of an observer at rest with the ground (i.e., in the

same frame as the barn), there is some characteristic speed vthresh, above which the ladder

will fit entirely within the barn. Calculate this characteristic speed.

Physics 4C Final Exam, Page 4 of 8 Spring 2020

(c) The results from parts (a) and (b) seem, at first glance, to be contradictory. In this and

the next part, we’ll attempt to rectify the situation, and ultimately show how the ladder’s

claim that it “never fits inside the barn” is perfectly compatible with the barn’s claim

that “the ladder does fit inside the barn” according to special relativity. We’ll start by

analyzing the situation using the Lorentz transformation. Denote the frame at rest with

the barn as the “unprimed” frame and the one at rest with the ladder as the “primed”

frame. We’ll assign the origin in both frames, (x, t) = (x

, t0

) = (0, 0), to be the event when

the leading end of the ladder reaches the barn’s exit. At the characteristic speed we found

in part (b), an observer at rest with respect to the barn would see the ladder just fit within

the barn (i.e., the trailing end of the ladder would be at the entrance to the barn at the

same instant in the unprimed frame). We denote this event by (x, t) = (−Lbarn, 0) in the

unprimed frame and by (x

, t0

) = (−Lladder, t0

) in the primed frame. Use these events with

the Lorentz transformation to determine t

0 and to verify the value of vthresh you calculated

in part (b).

(d) Based on the work done above, explain which set of observations (i.e., those made by the

observer on the ladder or the observer in the barn) is correct? If neither are correct or if

both are correct, state so clearly. Justify your answer in three or fewer sentences. In doing

so, be sure to explicitly state how the “contradiction” noted above is resolved.

Physics 4C Final Exam, Page 5 of 8 Spring 2020

3. Delta Function Potential: In the following questions, we’ll study the delta function potential,

a special case of the finite potential barrier analyzed in Section 7.6 of the course textbook. First,

we need to introduce the Dirac delta function (technically, its a generalized function, but no

need to worry about that). For our purposes, its best to take note of the following properties:

Z ∞

−∞

f(x)δ(x − a)dx = f(a) & δ(x − a) = 0 for x 6= a

Where δ(x − a) is the Dirac delta function, centered at x = a. Therefore, we can think of

δ(x − a) as a function that has a value of zero everywhere except at x = a, and whose integral

is equal to one so long as the limits of integration enclose x = a (and zero otherwise).

Based on the above properties, it should be clear that δ(x) has dimensions of [1/x]. So, in

order to write the delta function potential with appropriate dimensions, let us call the potential

energy function for our situation U(x) = λδ(x), where λ has dimensions of [energy · x]. For

simplicity, this places the delta function at x = 0, but our results will be easily generalizable.

For our analysis, we’ll be assuming λ > 0.

(a) Write the time-independent Schr¨odinger equation for the wave function ψ(x), valid for

−∞ < x < ∞. From the properties of the delta function noted above, it should be

apparent that if you break this up into two pieces, one to describe the “left” (−∞ < x < 0)

and one to describe the “right” (0 < x < ∞), it gets much simpler. Write these two pieces

down, using the subscripts of L and R to denote left and right wave functions.

(b) Under the assumption of a particle of energy E > 0 incident on the barrier from the left,

write down the wave functions ψL(x) and ψR(x). Given the stated assumption, you should

expect a total of three unknown constants to appear in your wave functions. [Note: the

differential equations you need to “solve” should look identical to ones we’ve solved already.

As such, you can just quote the result. If your equations don’t look familiar, or you aren’t

sure what the result is, you can ask me for this part.]

Physics 4C Final Exam, Page 6 of 8 Spring 2020

(c) We’re building up to a derivation of the reflection and transmission probabilities in part

(d), but before we can do that, we need to impose boundary conditions to relate the three

unknowns noted earlier. Normally these boundary conditions take the form of (i) requiring

continuity of ψ(x) at the origin and (ii) requiring continuity of ψ

(x) also at the origin.

While (i) works here, we’ll run into trouble if we try to apply (ii). We won’t have time to

derive this ourselves, but it turns out that we can use properties of the delta function to

obtain the following condition:

~

2

2m

[ψ

R(x = 0) − ψ

L

(x = 0)] = λψ(x = 0)

Apply (i) and the above expression to derive two constraints between the three unknowns

in your wave function.

(d) Finally, derive the reflection and transmission probabilities: R = |B/A|

2 and T = 1 − R.

Physics 4C Final Exam, Page 7 of 8 Spring 2020

Optional Extra Credit Problems

The problems below are optional and worth up to 5 points each.

1. Going back to your result for 1(b), consider a moment during which the object approaches

the focal point but does not quite reach it. You should find, according to your answer, that

the speed of the image grows without bound (i.e., it goes to infinity). Is this result correct?

Specifically, is it consistent with special relativity? If so, explain how, and if not, explain

where the error might have been introduced.

2. Return to the situation in problem 3. Explain in detail what the classical mechanics predictions for R and T would be for the case of λ > 0 (which we studied quantum mechanically)

and for the case of λ < 0. Return to the result we derived in 3(d), and use it to state the

quantum mechanical prediction for the case of λ < 0 (this shouldn’t require any extensive

re-derivation). Compare and contrast the classical- and quantum-mechanical predictions for

both possible signs of λ. Quantum mechanics is weird!

Closing Remarks

It has truly been my pleasure to have each of you in PHYS 4C this semester (and many of you in

PHYS 4A and 4B as well). Despite the very challenging circumstances that arose due to COVID-19,

you adapted and rose to new challenges, and now, you have persevered through the entirety of the

calculus-based “Physics for Scientists and Engineers” series that forms the basis of lower-division

Physics study. Take a moment to let that settle in, and pat yourself on the back. To reach the

point you find yourself in now is a significant accomplishment. If a STEM degree is in your future,

you are well on your way, and even if not, I hope the skills you learned in Physics will continue to

serve you as a critical thinker, informed citizen, and advocate for science. My heartfelt gratitude

for your patience and encouragement, both for me and for your peers, as we’ve navigated these

unprecedented times. Though I can’t do so in person, I hold out my hand to each of you in the

spirit of congratulations, respect, and to wish you farewell. Please stay in touch!

–Ben

Physics 4C Final Exam, Page 8 of 8 Spring 2020