Download C L 1 ~ R 2 C L S1 R C S2

1
Phys 202 Final Exam
E. Arık
Fall '03
Question 1 : Consider hydrogen atom with its electron being in a circular orbit of radius r . Calculate: (a) The total
energy E(r) . (b) The speed v(r) . (c) The angular momentum L, r and E as a function of n, by assuming 2πr = n
λdB , where λdB is electron’s de Broglie wavelength and n is a positive integer.
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Phys 202 Final Exam
E. Arık
Fall '03
Question 2 : The stopping potential for photoelectrons emitted from a surface illuminated by light of wavelength
500 nm is 0.7 V. When the incident wavelength is changed to a new value, the stopping potential is found to be
1.32 V. Calculate the new wavelength and the work function of the surface.
(take hc = 124 eV.nm)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Fall '03
Question 3 : A source containing a mixture of hydrogen and deuterium atoms emits light containing two closely
spaced red colors at λ = 656 nm whose separation is ∆λ = 0.2 nm. Find the minimum number of rulings needed in a
diffraction grating that can resolve these lines in the first order. Given that the slit separation d = 10λ, what is the
angular separation ∆θ and ∆λ at the first order. (sin θ ≈ tan θ ≈ θ)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Fall '03
Question 4 : The distance between the first and the fifth minima of a single slit diffraction pattern is 0.8 m. Given
that the screen distance D = 1.4 m . (a) Calculate the angle θ at which the first minimum occurs. (b) Calculate the
total number of maxima appearing on the screen. (sin θ ≈ tan θ ≈ θ)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Fall '03
Question 5 : A double slit arrangement with slit separation d, is illuminated with light of wavelength λ . Calculate
the full angular width (∆θ) at half maximum of an interference fringe. (sin θ ≈ tan θ ≈ θ)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Fall '03
Question 6 : A cylindrical wire of radius a, length L and resistivity ρ carries current i. Calculate the electric field
vector, the magnetic field vector at the surface of the cylinder and the rate at which the energy flows across the
surface of the cylinder.
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Phys 202 Final Exam
E. Arık
Fall '03
Question 7 : Consider the circuit shown in figure The AC generator supplies
150 V (max) with ω = 500 rad/s. With the switch open as shown in the figure,
the resulting current leads the generator emf by 45˚ . With the switch in position
1 the current lag the generator emf by 45˚. When the switch in position 2 the
maximum current is 3A . Find the values of R, L and C.
L C
C
1
~
2
R
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Fall '03
Question 8 : Consider the circuit shown in the figure . With switch S1 closed
and the other two switches open, the time constant is 1 s. With switch S2 closed
and the other two switches open, the time constant is 4 s. With switch S3 closed
and the other two switches open, calculate the period of oscillations.
S1
L
S2
S3
C
R
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
2
Phys 202 Final Exam
M. Mungan
Fall '05
Question 1 :
You have a cylindrical resistor of length l, radius a and resitivity ρ, such that the
resistance R is given by R = ρl/A. The resistor is carrying a current I .
(a) Show that the pointing vector S at the surface of the resistor is everywhere
directed normal to the surface as shown.
(b) The rate P at which energy flows into the resistor through its cylindrical
surface is calculated by integrating the pointing vector over this surface,
S
S
S
2a
i
i
S
S
S
l
P = ∫ S • dA .
(c) Find P as defined above and show that P = iR 2
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Phys 202 Final Exam
M. Mungan
Fall '05
Question 2 : A beam of light is mixture of horizontally polarized light. When it is sent through a polarizing sheet it
is found that the transmitted intensity can be varied by factor of 7 depending on the orientation of the polarizing
sheet. Calculate the ratio of the intensities Ihorizontal / Iunpolarized .
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
M. Mungan
Fall '05
A
B
n=1
Question 3 : Light of wavelength λ0 is incident on a soap film of unknown
thickness d. Find d given that the rays A and Bare out of phase by an amount
corresponding to (3/2)λ0. (see figure)
n=4/3
d
n=1
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Phys 202 Final Exam
M. Mungan
Fall '05
Question 4 : A single slits illuminated by two different wavelengths λ1 and λ2 . The first diffraction minimum of λ1
coincides with the second diffraction minimum of. λ2 . Find λ1/λ2.
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Phys 202 Final Exam
M. Mungan
Fall '05
Question 5 : An electron of mass m is confined to the region 0 ≤ x ≤ L. Calculate the frequency of the photon
emitted when the electron makes a transition from the n = 5 to the n = 3 state.
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Phys 202 Final Exam
M. Mungan
Fall '05
Question 6 : The potential energy of hydrogen atom can be written as U ( r ) = −
α ℏc
r
, where α ≈ 1/137 is
constant. Use Heisenberg’s uncertainty principle in form rp = ℏ to write the energy E as E(p) to find the ground
state (lowest) energy.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
M. Mungan
Fall '05
Question 7 : An ultraviolet light bulb, emitting at 400nm, and an infrared light bulb, emitting at 700nm , are rated
each at 130W.
a)which bulb radiates photons at greater rate?
b)how many more photons per seconds does it generate than the other bulb
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
M. Mungan
Fall '05
Question 8 Using up, down and strange quarks only, construct, if possible, a baryon
a)with total charge Q = +1 and strangeness S = -1
b)with Q = -1 and S = 0
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3
Phys 202 Final Exam
E. Arık
Spring '06
Question 1 : Consider hydrogen atom with its electron being in a circular orbit of radius r . Calculate:
(a) The total energy E(r)
(b) The speed v(r)
(c) The angular momentum L, r and E as a function of n, by assuming 2πr = n λdB , where λdB is electron’s de
Broglie wavelength and n is a positive integer.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Spring '06
Question 2 : The stopping potential for photoelectrons emitted from a surface illuminated by light of
wavelength 500 nm is 0.7 V. When the incident wavelength is changed to a new value, the stopping potential
is found to be 1.32 V. Calculate the new wavelength and the work function of the surface.
(take hc = 124 eV.nm)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Spring '06
Question 3 : A source containing a mixture of hydrogen and deuterium atoms emits light containing two
closely spaced red colors at λ = 656 nm whose separation is ∆λ = 0.2 nm. Find the minimum number of
rulings needed in a diffraction grating that can resolve these lines in the first order. Given that the slit
separation d = 10λ, what is the angular separation ∆θ and ∆λ at the first order. (sin θ ≈ tan θ ≈ θ)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Spring '06
Question 4 : The distance between the first and the fifth minima of a single slit diffraction pattern is 0.8 m. Given
that the screen distance D = 1.4 m,
(a) Calculate the angle θ at which the first minimum occurs,
(b) Calculate the total number of maxima appearing on the screen.
(sin θ ≈ tan θ ≈ θ)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Spring '06
Question 5 : The distance between the first and the fifth minima of a single slit diffraction pattern is 0.8 m. Given
that the screen distance D = 1.4 m,
(a) Calculate the angle θ at which the first minimum occurs,
(b) Calculate the total number of maxima appearing on the screen.
(sin θ ≈ tan θ ≈ θ)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Spring '06
Question 6 : A double slit arrangement with slit separation d, is illuminated with light of wavelength λ . Calculate
the full angular width (∆θ) at half maximum of an interference fringe. (sin θ ≈ tan θ ≈ θ)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Spring '06
Question 7 : A cylindrical wire of radius a, length L and resistivity ρ carries current i. Calculate the electric field
vector, the magnetic field vector at the surface of the cylinder and the rate at which the energy flows across the
surface of the cylinder.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final Exam
E. Arık
Spring '06
Question 8 : Consider the circuit shown in figure The AC generator supplies 150 V (max)
with ω = 500 rad/s. With the switch open as shown in the figure, the resulting current leads
the generator emf by 45˚ . With the switch in position 1 the current lag the generator emf
by 45˚. When the switch in position 2 the maximum current is 3A . Find the values of R, L
and C.
L
~
C
1
2
C
R
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
4
Phys 202 Final
M. Arık
SPRING 08
I
Question 1: A current I is flowing through a capacitor with circular plates of radius R.
Assuming that the charge distribution on the plates is uniform, calculate the magnetic field at a
distance r from center line of plates in the region between the plates.
R
r
B=?
I
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
M. Arık
SPRING 08
Question 2: The radius of the curved boundary is R. Find the location of the image
for an object at point P located a distance R from the curved boundary.
n n
n −n
a) By drawing the rays. b) By using the formula 1 + 2 = 2 1 .
p q
R
n2=2
n1=1
R
P
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
M. Arık
SPRING 08
Question 3: By using relativistic energy-momentum conservation prove that a photon cannot transfer all of its
energy to a free electron.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
M. Arık
SPRING 08
p 2 kx 2
+
. Assuming that p and
2m
2
x are operators satisfying px-xp=-iħ, show that H can be written as H=ħω(a*a+1/2) where ω is the
classical frequency and a, a* satisfy aa*-a*a=1. How should a and a* be defined in terms of x and p?
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Question 4: The energy of a harmonic oscillator is given by the Hamiltonian H =
Phys 202 Final
M. Arık
SPRING 08
Question 5: Particles of mass m, incident on a potential dip given by U(x)=0 for x<0 and U(x)=-U0 for x>0 are
described by the wave function ψ(x)= eikx for x<0. Find ψ(x) for x>0. Plot U(x).
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
M. Arık
SPRING 08
Question 6: According to classical physics a charge e moving with acceleration a radiates at a rate
dE
2 ke 2 2
a . a) Verify that the physical dimensions of both sides of this equation are the same. b) Show that
=−
dt
3 c2
dr
B
in classical hydrogen atom electron in circular orbit spirals towards the nucleus at a rate
= − 2 . c) Determine
dt
r
the constant B. Suppose electron starts from radius r0. Find the time it takes to fall to r=0 in terms of B and r0 .
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final
DEMIRALP, ERCAN, KAYA
SPRING 08
Question 3: What are the approximate dimensions of the smallest object on the Earth that astronauts can resolve by
eye when they are orbiting 250 km above the Earth? Assume λ=500nm and a pupil diameter of 5 mm.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final
DEMIRALP, ERCAN, KAYA
SPRING 08
Question 7: a) What is the energy of a photon of wavlength λ? b) What is the magnitude of momentum of a photon
of frequency f ? c) What is a black body ? d) Sketch intensity, I(λ, T), versus λ graph for a black body at
temperature T. e) What is the work function of a metal?
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
5
Phys 130 Final
Y. Skarlatos,
FALL 08
Question 7: Unpolarized light passes through two polarizing sheets and emerges from
the second with half its original intensity. Find the direction of polarization of the
second sheet relative to that of the first one.
I0
1/2
θ1
θ2
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final
Y. Skarlatos,
FALL 08
Question 8: At what temperature will the spectral radiancy of a cavity resonator be 16 times that at -173 0C?
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 121 Final
A.Sevgen
FALL 08
Question 5: A meter stick L0=1 meter is moving with a speed u=3/5 c with respect
to ground. What is its length L as measured in the ground frame of reference?
S
S`
u=3/5c
L0
x`=1m
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
M.Mungan
FALL 08
Question 1: Consider a double slit arrangement, where the slit separation is d and the distance from the
slit to the screen is L. A sheet of transparent plastic having index of refraction n and thickness t is placed
over the top slit. Assuming that the plastic is perfectly transparent,
a) By how much does the intereference pattern on the screen move?
b) Does it move upwards or downwards? Clearly explain your reasoning.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
M.Mungan
FALL 08
Question 2: At what speed v is the energy of a free particle twice its classical energy?
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
M.Mungan
FALL 08
Question 3: (Numerical) Neutrons travelling at a speed of 0.4 m/s are incident upon a double slit with slit
separation of 1 mm. An array of detectors is set up 10 m from the slit. How far off the axis is the first zero-intensity
point on the detector array?
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
M.Mungan
FALL 08
Question 4: Two light bulbs A and B are rated at the same power P, but λA < λB.
a) Which bulb radiates photons at a greater rate (photons/second)? Clearly explain why.
b) Calculate the rates at which each of the bulbs emits photons, express your answers in terms of the given
variables.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
M.Mungan
FALL 08
Question 5: (Numerical) Even the smoothest surfaces are "rough'' when viewed at a scale of 100 nm. When two
very smooth metal surfaces are placed in contact with each other, the actual distance between the surfaces varies
from 0 nm at points of contacts to about 100 nm. Take the average distance between the two surfaces due to
imperfect smoothness to be 50 nm. The work function φ of Al is 4.3 eV. What is the probability that an electron
will tunnel between two pieces of Al in contact? Express your answer in powers of 10, noting that ln 10~2.3.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
6
Phys 202 Final
FALL 08
M.Mungan
e2
Question 6: The potential energy of the hydrogen atom can be written as U ( r ) = −
4πε 0 r
a) Use Heisenberg's uncertainty principle in the form rp = ħ to write the energy E as E(p). b) Minimize E(p) to
find the ground state energy. c) (Numerical) Evaluate the expression found in b). Hint: might want to write the
potential energy in terms of α first.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
Spring '09
A. Sevgen
Question 1: Damped oscillations and Faraday : An RLC
series circuit is powered by a sinusoidal voltage source
R
C
V p (t ) = V0 cos ωt
ω0 2 =
a) Assume a direction for the current and show it on your
figure. Consider the closed loop line integral of the electric field
along the current direction assumed, and obtain the equation for
charge q on the capacitor.
b) If the driving frequency is equal to the resonant frequency of
the circuit, (ω = ω0 ) , and for the times much later than
V p (t )
~
1
LC
L
−1
R
 ) , compute the correlation function V p (t ) I p (t + ∆) .
L
1
1
(Hint: If Ap = Re. A0 eiωt and B p = Re.B0 eiωt , then Ap B p = Re. A0 B0* = Re. A00 B0 ( S xfl ) 2 )
2
2
damping time (t ≫ Td = 
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
Spring '09
A. Sevgen
Question 2: Radiation from an electric dipole array: Two
electric dipoles are situated as shown in the figure. Compute the
time-averaged intensity pattern of the radiation emitted in the
p1
direction rɵ = cos φ xɵ + sin φ ɵy , as given by the formula (2) given
below.
ɵ − iωt at the position a = ( −b , + a , 0) ,
p1 = p0 ze
1
2
2
ɵ − iωt at the position a = ( +b , − a , 0) ,
p2 = p0 ei∆ ze
2
2 2
 1  2 −i k i a1 new  ei ( kr −ωt ) 
E Total (r , t ) = E 1 (r , t ) + E 2 (r , t ) = 
p 0⊥ 

k e
4
πε
 r 

0 
(1) ,
new
O
y
rɵ
x
φ
i∆
p2 = p1e
L
where p 0 ⊥ = p0 zɵ ⊥ (1 + ei∆ e− ik r ⋅( a2 − a1 ) )
new 2
4
ck
p 0⊥
1 1 
The time averaged intensity is given by : I (rɵ ) = 2 
(2),

L  4πε 0 
8π
new 2 new 2
new 2
where p 0 ⊥ = p 0
- rɵ ⋅ p 0 and rɵ is a unit direction vector rɵ = cos φ xɵ + sin φ ɵy .
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
7
Phys 202 Final
A. Sevgen
Spring '09
y
Question 3: Photon Physics : A photon with polarization ψ = α x + β y
2
(with α + β
2
ψ
= 1 ) and moving in z-direction goes through a right-circular polarizer,
and then a vertical linear polarizer parallel to the y-axis.
RCP
a) What is the probability of photon going through this optical system ?
In other words you must compute p (ψ → R → y ) . Show that the initial photon state ψ
develops into a final
state ψ ... y . Find the probability amplitude ψ ... first, and then the probability.
i
π
1
1
e4
and β =
, compute probability numerically. Hint: R =
( x +i y ) .
b)For α =
2
2
2
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final A. Sevgen Spring '09
z
Question 4: Relativity : A sheet of charge with surface
density σ 0 is moving along the +x direction with speed u as
shown in the figure.
u
S
y
'
'
a)Find the fields E and B in S '
b)Find σ and js in S.
c)Find the fields E and B in S.
S'
x
σ ' = σ0
x'
L2 '
L1'
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final A. Sevgen Spring '09
∞
Question 5: Uncertainty principle and a simple quantum system: Use the
p2
uncertainty principle ∆x∆p x ≈ ℏ to estimate the ground state energy E = x
of a
2m
particle in an infinite square well potential of width L.
V
0
x
L
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
A. Sevgen
Spring '09
Question 6: Quantum: energy levels: One important observable for quantum systems is the energy operator. For a
two-level system, like the silver atoms, in the absence of any magnetic field, jhe energy operator is Eop = Eo 1op .
Therefore the energy of “up” atoms and “down” atoms are equal, E+ = E− = E0 . Thus up and down states z ±
are said to be degenerate. But if there is a magnetic field, then the energy operator becomes
Eop = E01op + J σ ⋅ B
(3)
where J is a real constant. For a uniform magnetic field in the z-direction, B = B0 zɵ show that the degeneracy is
lifted and compute the energy difference between the “up” atoms and “down” atoms, that is
∆E = E+ − E−
(4)
Where Eσ ' = zσ ' Eop zσ ' .
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
8
Phys 202 Final
A. Sevgen
Spring '09
Question 7: Quantum transformations: Consider the operator W
1
(5)
(σ x + σ z )
2
such that W W −1 = 1op . Write W −1 = Aσ x + Bσ z + Cσ xσ z + D1op and
W=
a) Compute the inverse operator W −1
determine the coefficients.
b) Compute W xσ ' =
1
x + +σ ' z −
(
2
)
c) Compute W −1 σ x W.
Hint: Propeties of Pauli matrices: σ a2 = 1op (a = x,y,z) and , σ aσ b = −σ bσ a = i
∑ε
abc
σ c (if a ≠ b )
(6)
c
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
A. Sevgen
Spring '09
Question 8: Quantum fluctuations : Suppose silver atoms are prepared in sme definite state a ' . We want to
compute the mean square deviation from the average value of the x-component of the angular momentum
1
S x = σ x , where σ x = − + + + − . Define the fluctuation operator S xfl = S x − S x .
2
2
a) Compute the coefficients A and B in ( S xfl )2 = A S x2 + B S x .
(Hint: First take the square of the operator S xfl , and then take the average.)
b)if a ' = z, + , compute the mean square deviation ( S xfl ) 2 .
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final A. Ankay , M. Mungan Spring 09
Question 3: A thin film of refractive index n and thickness t is put in front of one of the slits of a double-slit
experiment set-up. Find the angle θ by which the central maximum on a distant screen shifts relative to its location
without the thin film.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final A. Ankay , M. Mungan Spring 09
Question 4: A light beam is formed by superposing unpolarized light and polarized light. When the beam is passed
through a polarizer, it is found that the intensity of the beam can change by as much as a factor of 5 as the polarizer
is rotated(in other words I max = 5 I min ). Find the relative intensity of the two light sources.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final A. Ankay , M. Mungan Spring 09
Question 5: An electron is part of a quantum system. Assume that there are successive state transitions of the
electron, such that during the first transition a photon of frequency f1 is emitted which is followed by a transition
with emission of a photon of frequency f 2 . Show that there must also be a transition in which a photon of
frequency f= f1 + f 2 is emitted.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final A. Ankay , M. Mungan Spring 09
Question 6: A quantum particle of mass m is inside a one dimensional box of length L . Suppose that L is so small
p2
that the energy of the quantum particle has to be treated relativistically so that : E =
2m
is not valid anymore
(neither is the Schrödinger equation which is non-relativistic), but instead we have to use the relativistic expression
given by :
E = p 2 c 2 + m 2 c 4 Where c is the speed of light. It turns out that the wave function ψ ( x) is still
sinusoidal. Find the energy levels of the quantum particle.
9
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 L.Akant A.Ankay Final Spring '10
λ
θ1
Question 2 Coherent light rays of wavelenght λ strike a pair of narrow slits
separated by distance d at an angle θ1 as shown in Figure . On the other
side of this screen, at a great distance from it, there is another screen. The
region between the two screens is filled with a material of refraction index
n. Assume on the second screen an interference maximum is formed at an
angle θ 2 . Find the relation between θ1 , θ 2 , d , n and λ .
d
θ2
n
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 L.Akant A.Ankay Final Spring '10
Question 3 : A diffraction grating is illuminated with two light rays of wavelenghts λ and
3
λ . There are N slits
2
per unit lenght in the diffraction grating. Find the first two angles at which one observes overlapping bright fringes.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 L.Akant A.Ankay Final Spring '10
In a region of space, a quantum particle with zero energy has a wave function
Question 4 :
2
ψ ( x) = Axe −bx . Find the potential energy U as a function of x.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 L.Akant A.Ankay Final Spring '10
Question 8 : A free electron travels at a speed of 10-3c, where c is the speed of light. The electron is decelerated
and loses half of its kinetic energy by emitting a photon. If λ1 is the de Broglie wavelength of the electron after the
emission of the photon and λ2 is the wavelength of the photon, find the ratio
λ1
.
λ2
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
E.Oğuz
Spring '10
Question 1 : Consider the situation shown in figure. A light
ray of wavelength λ is incident on the second medium.
Calculate the minimum value of sinθ1 such that the ray is
observable in medium III (note n3 < n2 )
θ1
I
III
(n3 < n2 )
n1 = 1
n3
II
n2
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
E.Oğuz
Spring '10
Question 2 : Consider the double slit interference with light of wavelength λ. The slit separation is d. Slit-2 is
wider than slit-1 so that the light from slit-2 has an amplitude twice that of the light from slit-1 (ie E1 = E0 and E2 =
2 E0 ).
a. Calculate the intensity I(θ) on a distant screen as a function of θ (Ignore diffraction). b. Determine the positions
of the intensity maxima and minima. c. Plot I versus sinθ.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
E.Oğuz
Spring '10
Question 3 : Using the Lorentz transformation equations (x’ = γ(x-vt), t’= γ(t-vx/c2)) and suitable specific intervals
(ie using a relevant value for one or more of ∆x, ∆t, ∆x’ and ∆t’) prove the following:
a.
relativity of simultaneity,
b.
time dilation,
c.
length contraction.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
10
Phys 202 Final
E.Oğuz
Spring '10
Question 4 : Consider the Bohr theory of the Hydrogen atom.
a. State the assumption that leads to the quantization of the energy levels.
b. Derive the expression for energy levels.
c. State some of the main deficiencies of the Bohr theory.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
E.Oğuz
Spring '10
Question 5 : Consider the equilibrium in a cavity of temperature T. The density of normal modes (per unit volume)
is given by g(λ)dλ = (8π/ λ4 ) dλ.
a. Which physical quantity is given by by the expression (hc/ λ)(exp(hc/ λkBT)-1) -1 ?
b. Obtain the average energy density u(λ)dλ according to Quantum mechanics.
c. Prove that the total radiated intensity I is proportional to T4 .
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
E.Oğuz
Spring '10
Question 6 :
a. Explain clearly why the sky is blue.
b. Answer the following about the Greenhouse effect:
What causes it?
i.
ii.
Which molecules are (predominantly) involved?
iii.
What is the connection with the phenomenon of resonance?
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
E.Oğuz
Spring '10
Question 7 : Consider a quantum particle (of mass m) in an infinitely deep 1-d potential well of width L. Suppose
ψ(x,t) =
2 / L sin(πx/L)exp(-i(π2 ℏt/2mL2)). a) Calculate the average energy and average momentum of the
particle in this quantum state. b) Now suppose 2 / 3L sin(πx/L)exp(-i(π2 ℏt/2mL2)). +
i(4π2 ℏt/2mL2). Calculate the average energy of the particle in this quantum state.
4 / 3L sin(2πx/L)exp(-
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
E.Oğuz
Spring '10
Question 8 : The average function of the Hydrogen atom in its ground state is
ψ( r ) = (1/ πa03 )exp(-r/a0) where a0 is the Bohr radius. Find
a. The probability P (r )d 3r that the electron will be found in the volume d 3r around r ,
b. The probability Pdr that the electron will be found within the infinitesimal spherical shell of
radius r and thickness dr.
c. Calculate the rms uncertainity
(r − r
2
.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
A. Sevgen
Fall '10
Question 1 : Forced and damped oscillation:
Consider a series RLC circuit in the shape of a rectangular loop only partly
in a B -field which is oscillating with frequency ω. The loop extends a
distance x into the B -field and is held fixed in position. Take B=B˳sin ωt
with B>0.
a) Write the line integral of the electric field along the direction of the
current shown in the figure. And then obtain the differential
R
B0sin(ωt)
L
l
I
C
x
11
dφ
equation obeyed by the current I. (Hint: E
∫ .dl = − dt . Note that the flux ϕ is due to i) the magnetic field
due to the current in the circuit, and ii) the external magnetic field).
b) Solve the equation you found in part-a for the steady state current if the external magnetic field is driven at
the resonance frequency ω˳ = ( LC ) −1
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
A. Sevgen
Fall '10
y
p1
Question 3 : Radiation from an electric dipole
array: Two electric dipoles are situated as
shown in Figure.1 below:
p2
r
a/2
ϕ
x
a/2
L
Figure 1: Two dipoles placed along x-axis and parallel to z-axis
a) Compute I ( rˆ)
b) What are the maximum and minimum values of I ( rˆ) ?
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
A. Sevgen
Fall '10
Question 3 : Uncertainty principle and the hydrogen atom: In this problem you will show that Hydrogen atom
is the balance between the Coulomb attraction and the outward quantum pressure and that the Uncertainty principle
makes it certain that atoms are stable.
a) The Energy of the hydrogen atom is sum of the kinetic and potential energies:
E=
p
1 e2
−
2m 4πε 0 r
Use the uncertainty principle, in an approximate way, ∆p∆r≈ ℏ , and p≈∆p and also r≈∆r
to find the energy as a function of r, that is find E(r). Make a rough sketch of E(r) as a function of r.
b) The distance r˳ for which the energy is minimum is found from
dE
(r˳) =0, which gives the size of the
dr
Hydrogen atom as r˳. What is the expression for r˳ and estimate it numerically.
(Hint: mc 2 =
1
e2 1
1
MeV ,1MeV = 106 , hc ≈ 2000eVA0 ,
≈
)
2
4πε 0 ℏc 140
c) Find the expression for the binding energy of the Hydrogen atom E(r˳), and estimate it numerically. How
does it compare to the mean kinetic energy of air particles near temperature which is about 1/40eV? Thus by
this comparison, is hydrogen strongly or weakly bound at room temperatures?
12
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
A. Sevgen
Fall '10
Question 4 : Relativity – kinematics: [A,B] is the platform of a S
A’
B’
O’
train station of length L˳ in S-ground frame. A train with end points
u
Train
marked A’ and B’ and passing with speed u (such that γ(u)=2) is
observed to line up with the ends of the station platform
O
A
B
simultaneously. That is B 'on B and A 'on A are simultaneous events
in the S-station frame. O and O’ are fixed observers on the platform
L0
and on the train respectively.
a) What is the time necessary for the train to pass the observer I on the platform?
b) What is the rest length L˳’ of the train?
c) According to the observer O’ on the train what is the length D of the station platform?
d) According to the observer O’ on the train, how long does it take for O to pass the entire length of the train?
e) According to the observer O on the platform B 'on B and A 'on A are simultaneous, but not for O’ on the
train. When Aon A ' in the train frame, where is B, and how much time has passed since Bon B ' ?
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
A. Sevgen
Fall '10
Question 5 : Relativity – charges, currents, fields
A sheet of charge with surface charge density σ˳ is moving
along the +x-direction with speed u as shown in the figure:
S
a) Find the fields E ’ and B ’ in S’.
b) Find σ and js in S.
c) Find the fields E and B in S.
S’
σ’=σ˳
y
y’
x’
x
L1
L2
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
A. Sevgen
Fall '10
Question 6 : Quantum: probability and probability amplitude: A three level atom is prepared in the state <a’|.
A measurement of property B can give the three possible results b1 , b2 , b3 .
a) Write the expression for the probability process p(a’ → ( b1 ), ( b2 OR b3 ) → c’)
where we know the intermediate states were b1 , and b2 OR b3 , but we don’t know if the system
went through b2 or b3 .
b) Write the expression for the probability of the process p(a’ → ( b1 ), b2 , b3 → c’)
where we measure all the intermediate states b1 , b2 or b3 , but make no selection.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
A. Sevgen
Fall '10
Question 7: Quantum: transformations Consider the operator W =
1
(σ x + σ z )
2
a) Use the properties of Pauli matrices and computer W 2
b) Computer W|z+>
c) Computer σ x ( W|z+>). Thus interpret the state W|z+>.
Hint: Properties of Pauli matrices: σ a2 = 1op (a=x,y,z) and, σ aσ b = −σ bσ a =i
∑∈
abc
σ c (if a ≠ b)
c
σ x =| −+ | + | +− | , σ y = i (| −+ | − | +− |) and σ x =| ++ | − | −− | .
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
13
Phys 202 Final
A. Sevgen
Fall '10
z
Question 8: Quantum: transformations We know that for a two-level magnetic system
we have the eigenvalue equations
σ z | zˆσ ' >= σ ' | zˆσ ' > where σ’ = +1, -1
(1)
If we rotate the coordinate system around the y axis by angle β, we know from
Homework-9 that,
−1
σ z =R σ x R =σ z cos β + σ x sin β
You can easily obtain the eigenvectors of
where R = e
iβ
z
β
y
x
β
σy
2
σ z as follows: Multiply the Equation 1 from
x
the left R −1 and see that you can bring it to the form σ z | zˆσ ' >= σ ' | zˆσ ' >
where you will also find out the definitions of the vectors | zˆσ ' > . Write out explicitly the eigenvectors | zˆ + > and
| zˆ − > in terms of the original basis set { | zˆ + > and | zˆ − > }.
(Hint: You may find the following relations helpful: e
iβ
σy
2
= 1op cos β + iσ y sin β
,
iσ y =| +− | − | −+ |
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final
M. Arık
Spring '11
3.
Two slits seperated by d emit light of wavelength λ in the same phase but different amplitudes
E1 = 3 E0 cos(ωt)
E2 = 4 E0 cos(ωt)
When light is observed at angle θ from the center line a phase difference δ is present between the two waves.
a) Write δ in terms of d, θ and λ. (2 pts)
b) Calculate the intensity as a function of θ. (3 pts)
c) Calculate the ratio of of the maximum and the minimum intensities. (2 pts)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final
M. Arık
Spring '11
4. Assume light of wavelength 600 nm passes through two slits 3.00 µm wide, with their centers 9.00 µm apart.
Make a sketch of the combined diffraction and interference pattern in the form of a graph of intensity versus ϕ =
(πa sinθ)/λ. Carefully label the ϕ – axis.
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 130 Final
M. Arık
Spring '11
5.
A photon having wavelength λ scatters
off a free electron at A (as in figure),
producing a second photon having
wavelength λ′. This photon then scatters off
another free electron at B, producing a third
photon having wavelength λ′′ and moving in
a direction directly opposite the original
photon as shown in the figure. Determine
the value of ∆λ = λ′′ - λ. You may use the
Compton scattering formula
λ′- λ = λC (1 – cos θ).
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
14
Phys 130 Final
M. Arık
Spring '11
6. A particle of mass m and energy E<U0 is under the influence of a potential in one dimension given by U = U0 for
0 < x < L and U = 0 otherwise. In the region 0 < x < L the wave function is given by ψ = exp(k′x) where ћk′ =
(2m(U0 – E))½, also take ћk = (2m E)½ .
a) Draw and state what this wave function represents in terms of tunneling from which region into which region? (1
pt)
b) Write the wave function in region x < 0 in terms of amplitude A1 and phase ϕ1. (1 pt)
c) Write the wave function in region x > L in terms of amplitude A2 and phase ϕ2. (1 pt)
d) Find the four equations obtained by imposing the condition that the wave function and its derivative are
continuous at x = 0 and at x = L. (2 pts)
e) Eliminate ϕ1 and ϕ2 from the four equations and show that the transmission coefficient is exp(-2k′L). (2 pts)
₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪₪
Phys 202 Final
??
Spring '11