© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
Physics 322 Final Exam Study Guide
Date of Exam: Tuesday, May 12, 2:00 pm - 4:00 pm
Rules:
1. Closed Book and Closed Note. A formula sheet will be provided.
2. You will be allowed the entire 2 ½ hours for the exam, although I will be aiming for it
to be 90 minutes in length.
3. You will be allowed a calculator (although I will avoid problems where it will be
helpful).
4. If you are stuck because you can’t remember a formula or an indefinite integral, I will
assure you that you can ask me… I am not big on formula memorization.
5. Combat Arms is not to be played until after the final exam is over.
Topics on this Exam:
This final exam is cumulative, but focused on more recent material. As a warning,
the problems that you all had difficulty with on previous exams and homework
might may a comeback in some form. In addition to material covered in previous
exams, you can expect the problems on the final exam to cover the concepts in Chapter 7,
Chapter 8 (Sections 1 through 4), and Chapter 11 (Sections 1 through 3, 5, and 6) of
Harris’ Modern Physics textbook and problem sets 9 through 12.
BIG HINTS:
•
•
•
Understand the answers to the problem sets you did – some problems on the exam
will be very similar to them.
Understand the examples in the textbook.
By now you have noticed that on the midterms, I avoid problems with particularly
nasty integrals/derivatives, focusing instead on mathematically simpler problems
that test your conceptual knowledge and ability to apply those concepts. This
same approach holds for the problems I will be writing for the final exam.
Chapter 7) Quantum Mechanics in Three Dimensions and the Hydrogen
Atom
1. The Schrödinger Equation in 3 Dimensions
a. Know the general changes that have to go on in order to get our previously
1 dimensional Schrödinger equations to work in three dimensions. I don’t
mean memorizing the newer equations, I mean understanding just what
needed to be modified in moving to 3 dimensions.
2. The 3D Infinite Well
a. Understand how the technique of separation of variables is used to
determine the allowed wave functions and states for a 3 dimensional
–– Page 1 of 11 ––
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
infinite well.
b. Know under what situations a degeneracy in allowed energy states occurs
and more importantly what it means.
c. Why are there three quantum numbers instead of just one for a 3-D infinite
well? What does this mean physically?
d. When will degenerate energy levels split? How might it affect the allowed
photon wavelengths due to electron transitions between different energy
states?
3. Energy Quantization and Spectral Lines in Hydrogen
a. Know the basic characteristics of the hydrogen spectra and why it strongly
suggests that the allowed energy levels of the electron in the hydrogen
atom are quantized.
b. Given the expression for the energy levels of the hydrogen atom, be able to
compute the wavelength of a photon emitted/absorbed by an electron to
accomplish a transition between any two energy levels?
4. The Schrödinger Equation for a Central Force
a. Understand the spherical coordinate system used by Harris (in addition to
the radial coordinate r, it takes θ as the “polar angle” which measures
angle from the “north pole” of the coordinate system and ϕ as the
azimuthal angle which measures angle about the z-axis from the x-axis).
b. Understand what is meant by the separability of the wave function into
three wave functions in each coordinate:
ψ ( r,θ , φ ) = R ( r ) Θ (θ ) Φ (φ )
c. Examine “A Closer Look” in p. 247 and understand how the separability
of the wave function allows use to find three equations: the radial, the
polar, and the azimuthal equations which when solved let us solve for
R ( r ) , Θ (θ ) , and Φ (φ ) respectively. You don’t need to memorize the
equations, I just want you to understand what they represent.
5. Angular Behavior in a Central Force
a. Understand how the solution to the azimuthal equation Φ (φ ) is obtained
and why the solution must be sinusoidal in nature (HINT: Smoothness
and boundary conditions).
b. Know what the solutions to the azimuthal equation Φ (φ ) look like (such
as those shown in Figure 7.9 on p. 249).
c. Why does he azimuthal equation Φ (φ ) lead to the quantization of the zcomponent of the orbital angular momentum, Lz in terms of (magnetic)
quantum number mℓ ?
–– Page 2 of 11 ––
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
d. Understand conceptually how we arrive at the quantization of the
magnitude of the orbital angular momentum |L| via the solution of the
polar equation.
e. Understand how the orbital angular momentum |L| and the orbital
quantum number ℓ are related.
f. Operationally, what limits does the orbital quantum number ℓ place on the
allowed values of (magnetic) quantum number mℓ and why does this
physically make sense?
g. How do the quantum numbers ℓ and mℓ relate to the orbital motion of the
electron (as illustrated in Figures 7.13 and 7.14)?
6. The Hydrogen Atom
a. Know conceptually how the radial solutions, as shown in Table 7.4, are
obtained (even if actually solving for them is beyond the scope of this
course).
b. How does the principal quantum number n restrict the values on the
quantum numbers ℓ and mℓ ?
c. Know why as the principal quantum number n increases there is an
increasing degeneracy in the energy levels of the hydrogen atom (HINT:
Examine Table 7.5).
d. Know why it is we can normalize the radial and angular components of
the hydrogen atom wave equation independently.
e. Annoyingly, you should probably memorize the order of the letters used in
spectroscopic notation to indicate the various ℓ states (NOTE: The order
is s, p, d, f, g, h,...)
7. Radial Probability
a. Understand what is represented by the radial probability in equation 7-38
(a probability per unit radial distance).
b. Know the difference between most likely location, most likely radius, and
average radius (aka the expectation value of the radius) and how to
compute those three values. A few of you really blew this on problem set
10...
8. The Hydrogen-like Atom
a. Basically, know how you have to modify the hydrogen atom model (it’s
only the potential that changes) when dealing with hydrogen-like atoms
which are atoms with Z protons and one electron.
–– Page 3 of 11 ––
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
Chapter 8) Spin and Atomic Physics [Sections 1 to 4]
1. Evidence of Angular Momentum Quantization: A New Property
a. Know how the magnetic dipole moment is related to angular momentum.
b. Know how the Stern-Gerlach apparatus was designed to separate a beam
of particles fired through it by their magnetic dipole moment. Specifically
how does it manage to exert a force (as opposed to a torque) on the
dipoles.
c. How did firing ground state hydrogen atoms through the Stern-Gerlach
apparatus reveal the existence of an intrinsic angular momentum (called
spin) in the electron?
d. Why does the spin of the proton in the hydrogen atom not play a huge role
in the Stern-Gerlach experiments? HINT: Look at the proper relativistic
quantum mechanical expression for the magnetic dipole moment, equation
8-7.
e. What is meant by the statement “the electron is a spin-½ particle”?
f. What is different about the dimensionless number s used to quantize spin
and other quantum numbers?
2. Identical Particles
a. Understand what is meant by “identical particles”.
b. Understand the setup of the separable “Two Particles in a Box” problem
(detailed in “A Closer Look” in p. 301) used to solve for the expression of
two-particle wave function for two particles in an infinite one-dimensional
well.
c. Why must identical particles be indistinguishable and why does that mean
we must have a “symmetric” probability density?
d. What do the symmetric and antisymmetric two-particle wave functions
represent?
e. What are fermions? What are bosons?
3. The Exclusion Principle
a. Know what is meant by the (Pauli) Exclusion Principle.
b. Why does the Exclusion Principle apply to fermions but not bosons?
4. Multi-electron Atoms and the Periodic Table
a. Know the rule of thumb for the order of filling electron shells and
subshells, the “ n + ℓ ” rule.
b. Know what electronic configuration notation indicates (e.g. – 1s2 or
2s22p63s23p1).
c. What makes the periodic table of the elements periodic?
d. Which electrons (in any given atom) are the ones that govern the
chemistry of atoms?
–– Page 4 of 11 ––
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
Chapter 11) Nuclear Physics
1. Basic Structure
a. Know the basic definitions of: nucleon, atomic number (Z), mass number
(A), and isotopes.
b. What is the relationship between radius of a nucleus and the number of
nucleons and HOW was this determined?
2. Binding
a. What is the “strong nuclear force” and what do we know about it?
b. What role does the fact the strong nuclear force only has a range of ~1 fm
have on the nature of internucleon bonds (compared to say electrostatic
forces).
c. What is an atomic mass unit (u)? I don’t mean just knowing to look up
unit conversion, I mean knowing that 1 u = 1 gm/mol.
d. What is meant by “binding energy” and why is it we can say a nucleus
with a high binding energy has a very low energy?
e. What role do the following play in determining the binding energy of the
nucleus? The strong internucleon attraction. Coulomb repulsion. The
exclusion principle.
f. Explain why the binding energy is really just the energy equivalent of “the
mass of the parts” minus “the mass of the whole”? How can there be a
difference between the mass of the parts that make up the whole and the
whole nucleus?
g. What is the curve of stability (Figure 11.13) and why does it exist?
h. Why do we care about the binding energy per nucleon as a function of Z?
i. Understand how to use the data in Appendix I if it is given to you!
3. Nuclear Models
a. What was the motivation for the liquid drop model of the nucleus?
b. Know why the various terms (the volume term, the surface term, the
Coulomb term, and the asymmetry term) in the liquid drop model relate to
physically and why.
c. How were the terms in the semi-empirical binding energy formula
determined?
d. Why did we also (briefly) consider a second model for the nucleus of the
atom called the shell model? What does the shell model allow us to
explain that the liquid drop model doesn’t?
e. What are magic numbers? How is their explanation under the shell model
similar to the concept of “noble” gases on the periodic table? NOTE:
Don’t memorize the numbers, learn the concept.
–– Page 5 of 11 ––
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
4. Radioactivity
a. What is meant by the term “radioactive decay”?
b. What drives every form of radioactive decay?
c. What is Q? (I mean conceptually) What values of Q are associated with
radioactive decay processes?
d. What is a parent nucleus and what is a daughter nucleus?
e. What is α decay? What is β– decay? What is β+ decay? What is electron
capture? What is γ decay?
f. How did observations of the kinetic energies of β– particles in β– decay
result in the discovery/identification of the neutrino?
g. Understand why Qβ − = m parent − mdaughter c 2 if those are atomic masses?
h. Understand why Qβ +
(
= (m
)
parent
)
− mdaughter − 2me c 2 if those are atomic
masses?
i. Understand why QEC = m parent − mdaughter c 2 if those are atomic masses?
(
)
j. What is spontaneous fission and how is it distinct from other forms of
radioactive decay?
k. What happens in the nucleus (in terms of changes in Z and N) during α
decay? during β– decay? during β+ decay? during electron capture? during
γ decay? during spontaneous fission?
5. Radioactive Decay
a. Understand how the idea that the number of nuclei decaying per unit time
dN
(aka the decay rate) being proportional to the number of nuclei (
∝N )
dt
leads to an exponential decline in the number of nuclei over time.
b. What is the definition of decay constant? half-life? How are the two
related?
c. Be able to compute how much of an original radioactive isotope is
available after n half-lives have passed.
d. How does radioactive dating work? What information do we need to have
to make it work?
–– Page 6 of 11 ––
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
Physical Constants for Final Exam (First Draft)
1eV = 1.602 × 10 −19 J
e = 1.602 × 10 −19 C
c = 3.00 × 10 8
m
s
h = 6.626 × 10 −34 J i s
h
!=
= 1.055 × 10 −34 J i s
2π
J
kB = 1.38 × 10 −23
K
1
2
= 8.99 × 10 9 NC·m2
4π ε0
me = 9.1094 × 10 −31 kg
m p = 1.6726 × 10 −27 kg
mn = 1.6749 × 10 −27 kg
1u = 1.6605 × 10 −27 kg = 931.5
mH = 1.007825u
MeV
c2
mn = 1.008665u
m p = 1.007276u
me = 0.000549u
–– Page 7 of 11 ––
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
Equations for Final Exam (First Draft)
Special Relativity (Repeats from Midterm #1)
Implications of Einstein’s Postulates
1
γv =
v2
1− 2
c
Δt = γ v Δt 0
L
L= 0
γv
Energy and Momentum
Lorentz Transformations
Velocity Transformations
x ' = γ v ( x − vt )
x = γ v ( x '+ vt ')
⎛ v
⎞
t ' = γ v ⎜ − 2 x + t⎟
⎝ c
⎠
⎛ v
⎞
t = γ v ⎜ + 2 x '+ t '⎟
⎝ c
⎠
1
γu =
1−
u2
c2
KE = (γ u − 1) mc 2
!
!
p = γ u mu
E = γ u mc 2
Einternal = mc 2
ux ' =
E 2 = p2c2 + m 2c4
uy
uz
ux − v
, uy ' =
, uz ' =
ux v
ux v
uv
1− 2
1− 2
1 − x2
c
c
c
Doppler Shift
fobs = fsource
1−
v2
c2
v
1 + cosθ
c
Waves as Particles and Particles as Waves (Repeats from Midterm #1)
Properties of Light Waves/Particles
λf = c
hc
E = hf =
= pc
λ
Properties of Matter Waves/Particles
E = hf = !ω
p=
h
= !k
λ
probability density ∝ Ψ ( x,t )
2
Photoelectric Effect / Compton Effect
KEmax = hf − φ
h
λ′ − λ =
(1 − cosθ )
me c
Free-Particle Schrödinger Equation (1-D)
∂Ψ ( x,t )
−! ∂ 2 Ψ ( x,t )
= i!
2
2m ∂x
∂t
Heisenberg Uncertainty Principle
!
ΔxΔpx ≥
2
!
ΔEΔt ≥
2
Bohr Atom
mvr = n!
4π ε0 ) ! 2 2
(
r=
n
E=−
–– Page 8 of 11 ––
me2
me4
2 ( 4π ε0 )
2
1
1
= −13.6eV 2
2
n
! n
2
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
Quantum Mechanics: 1-D Bound States (Repeats from Midterm #2)
Time-Dependent Schrödinger Equation
−
∂Ψ ( x,t )
! ∂ Ψ ( x,t )
+ U ( x ) Ψ ( x,t ) = i!
2
2m ∂x
∂t
2
2
Ψ ( x,t ) dx = 1
∫
2
all space
Time-Independent Schrödinger Equation
Ψ ( x,t ) = ψ ( x )φ ( t )
where φ ( t ) = e
E
−i t
!
! 2 d 2ψ ( x )
+ U ( x )ψ ( x ) = Eψ ( x )
2m dx 2
Infinite Well - Particle in a Box
−
⎧⎪ 0,
0<x<L
U ( x) = ⎨
⎩⎪ ∞, x < 0 or x > L
⎧ 2
⎛ nπ x ⎞
⎪
sin ⎜
,
0<x<L
ψ ( x) = ⎨ L
⎝ L ⎟⎠
⎪
0,
x < 0 or x > L
⎩
n 2π 2 ! 2
En =
2mL2
General Solutions
2m ( E − U 0 )
!2
ψ ( x ) = Ae+ikx + De−ikx
k=
2m (U 0 − E )
!2
ψ ( x ) = Ce+α x + De−α x
α=
Simple Harmonic Oscillator Potential
⎛
En = ⎜ n +
⎝
1⎞
⎟ !ω 0 where n = 0,1, 2, 3,... and ω 0 =
2⎠
κ
m
Operators
xˆ = x
∂
∂x
∂
Eˆ = i!
∂t
pˆ = −i!
Q=
∫
ˆ ( x,t ) dx
Ψ * ( x,t ) QΨ
all space
ΔQ = Q 2 − Q 2
Quantum Mechanics: Unbound States (Repeats from Midterm #2)
Potential Steps for Stationary States
2
2
ψ inc = ψ ref + ψ trans
2
ψ
k
T = trans 2 trans
ψ inc kinc
R=
For E > U 0 :
T=
(
(
R=
(
4 E ( E − U0 )
E + E − U0
E − E − U0
E + E − U0
Quantum Tunneling – Wide-Barriers
2
ψ ref
2
ψ inc
2
kref
kinc
2m (U 0 − E )
L
= αL =
≫1
δ
!
E ⎛
E ⎞ −2 δL
T ≅ 16 ⎜ 1 − ⎟ e
U0 ⎝ U0 ⎠
Potential Barriers for Stationary States
)
)
)
2
Quantum Tunneling – For E < U 0 :
E ⎛
E⎞
1− ⎟
⎜
U0 ⎝ U0 ⎠
2m (U 0 − E ) L ⎤
E ⎛
E⎞
⎥ + 4 ⎜ 1− ⎟
!
U0 ⎝ U0 ⎠
⎥⎦
4
2
2
T=
⎡
sinh 2 ⎢
⎢⎣
R = 1− T
–– Page 9 of 11 ––
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
Quantum Mechanics: 3-D Bound States (Repeats from Midterm #2)
Time-Dependent Schrödinger Equation
∂2
∂2
∂2
∇2 = 2 + 2 + 2
∂x
∂y
∂z
!2 2 "
"
"
"
∇ Ψ ( r,t ) + U ( r ) Ψ ( r,t ) = i!Ψ ( r,t )
2m
" 2
∫ Ψ ( r,t ) dV = 1
−
Time-Independent Schrödinger Equation
!2 2 "
"
"
"
−
∇ ψ ( r ) + U ( r )ψ ( r ) = Eψ ( r )
2m
" 2
∫ ψ ( r ) dV = 1
all space
all space
3-D Infinite Well
⎛ 2⎞
ψ nx ,ny ,nz ( x, y, z ) = ⎜ ⎟
⎝ L⎠
Enx ,ny ,nz =
3/2
⎛ n π x ⎞ ⎛ n yπ y ⎞ ⎛ n z π z ⎞
sin ⎜ x ⎟ sin ⎜
⎟ sin
⎝ Lx ⎠ ⎝ Ly ⎠ ⎜⎝ Lz ⎟⎠
2
π 2 ! 2 ⎛ nx2 ny nz2 ⎞
+
+
2m ⎜⎝ L2x L2y L2z ⎟⎠
Quantum Mechanics: Central Forces and The Hydrogen Atom
Quantized Quantities in Hydrogen Atom
En = −
me 4
1
2
2
2 ( 4π ε0 ) ! 2 n
where n = 1, 2, 3,...
L = ℓ ( ℓ + 1)!
where ℓ = 0,1, 2, 3,..., n − 1
L z = mℓ !
where mℓ = 0,1, 2, 3,..., ℓ
Normalization
Hydrogen Atom
π
2π
4π ε0 ) ! 2
(
2
2
2
∫ R ( r ) r dr ∫θ = 0 ∫φ = 0 Θ (θ ) sinθ dθ dφ = 1 a0 = me2 = 0.0529nm
r=0
∞
or split up
∞
∫
R 2 ( r ) r 2 dr = 1
r=0
π
∫ ∫
2π
θ =0 φ=0
Radial Probability
P ( r ) = r 2 R2 ( r )
Θ 2 (θ ) sin θ dθ dφ = 1
–– Page 10 of 11 ––
© Copyright 2015 Juan E. Cabanela. All Rights Reserved.
Duplication only permitted for students in my Physics 322 course.
Quantum Mechanics: Spin and Atomic Physics
Magnetic Dipole Moment
!!"
e "
µL = −
L
2m
" " "
τ =µ×B
" " "
"
F = −∇ − µ ·B
(
#
Spin
S = s ( s + 1)!
"#
"
q #
µs = g
S
2m
Sz = ms ! where ms = −s, −s + 1,..., s − 1, s
)
Identical Particles
for B varying
only in z direction
ψ S ( x1 , x2 ) ≡ ψ n ( x1 )ψ n ' ( x2 ) + ψ n ' ( x1 )ψ n ( x2 )
"
∂B
∂B
e
F = µ z z zˆ = −
mℓ % z zˆ
∂z
2m
∂z
[Symmetric]
ψ A ( x1 , x2 ) ≡ ψ n ( x1 )ψ n ' ( x2 ) − ψ n ' ( x1 )ψ n ( x2 )
[Antisymmetric]
Nuclear Physics
Size
Radioactivity
(
r = A × R0 where R0 = 1.2fm
)
Qβ +
BE = ZmH + Nmn − M A X c 2
Z
(
= (m
= (m
QEC
Liquid Drop Model
− mdaughter − 2me c 2
parent
− mdaughter c 2
Radioactive Decay
Z ( Z − 1)
(N − Z)
2/3
BE = c!
− c3
− c4
1A − c
2A
1/3
!
A $% "$#A$%
"$#
Volume
Surface
Term
BEMeV = 15.8A − 17.8A
Coulomb
Term
2/3
Asymmetry
Term
Z ( Z − 1)
(N − Z)
− 0.71
− 23.7
1/3
A
A
2
–– Page 11 of 11 ––
)
parent
2
Term
)
Qβ − = m parent − mdaughter c 2
Binding Energy (Experimental)
(
)
Q = mi − m f c 2
1/3
dN
= − λ N = −R
dt
N = N 0 e− λt
N0
= N 0 e− λT1/2
2
ln 2
λ=
T1/2
)