Phys 2601 Final Exam
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Final Exam
Phys 2601
1 Useful Formulae
Constants and unit conversions
h = 6.63 x 10 —34 Js = 4.14 x 10 — 15 eVs
方 = = 1.06 x 10 —34 Js = 6.58 x 10 — 16 eVs
c = 3.0 x 108 m/s
1 eV = 1.602 x 10 — 19 J
e = 1.60 x 10 — 19 C
me = 9.11 x 10 —31 kg = 0.511 MeV/c2
me /方2 = 1.31 x 1019 ( eV m2 ) — 1
kB = 1.38 x 10 —23 m2 kg s —2 K — 1 (Boltzmann’s constant) σ = 5.67 x 10 —8 W / (m2 K4 ) (Stefan’s constant)
λC = = 2.43 x 10 — 12 m (for an electron)
µb = 9.2740100783 x 10 —24 J/ T
α =
a0 = 0.529 A
Useful formulae
eix = cos(x) + i sin(x) , cos(x) = ╱eix + e —ix 、, sin(x) = ╱eix _ e —ix、
General:
RT = σT4 ρT (ν)dν = dν KEmax = eV0 E = hν E = pc (m0 = 0) λdB = h/p ∆λ = λC (1 _ cos θ) Intensity = ∈2 = Nhν ∆x∆px 2 2(方) κ = 1/λ dν En = _ ╱ 、2 2(d)方(κ)2(1)n2 Hydgen _ |
λmax T = 2.898 x 10 —3 m K
KEmax = hν _ w0 E = |p2 c2 + m0(2)c4 p = mv/^1 _ v2 /c2 λ = c/ν (electromagnetic radiation) nλ = d sin θ
∆E∆t 2 2(方) |
Schroedinger Equation in 1 dimension:
pˆ = _i方∂/∂x
k = |2m(E _ V0 )/方 , κ = |2m(V0 _ E)/方
i方∂t Ψ(x, t) = _2m(方2)∂x(2)Ψ(x, t) + V (x)Ψ(x, t)
Eψ(x) = _2m(方2)∂x(2)ψ(x) + V (x)ψ(x)
Schroedinger Equation for Hydrogen-like atoms
_2(方)2µ ┌ ╱r2 、 + ╱sin θ、 + ┐ ψ(r, θ, φ) _ ψ(r, θ, φ) = Eψ(r, θ, φ)
ψn,l,ml = Rn,l (r)Ylml (θ, φ)
2 = _方2 ┌ ╱sin θ、 + ┐
L2 = 方2 l(l + 1)
x = i方 ╱sin φ + cot θ cos φ、
y = i方 ╱ _ cos φ + cot θ sin φ、
z = _i方
0 < l < n , ml = _l, _l + 1, ...l _ 1, l
Pnl (r)dr = |Rnl (r)|2 (4πr2 )dr
Perturbations to hydrogen-like atoms
∆E = _µ . B = µb (ml + 2ms )B , µz = µb /方(z + 2 z )
1 1 S . L
2m2 c2 4π∈0 r3
j = l + 1/2, l _ 1/2
S . L = (J2 _ L2 _ S2 ) =2(方2) (j(j + 1) _ l(l + 1) _ s(s + 1)) ψS = [ψa (r(1), ms (1))ψb (r(2), ms (2)) + ψb (r(1), ms (1))ψa (r(2), ms (2))] ψA = [ψa (r(1), ms (1))ψb (r(2), ms (2)) _ ψb (r(1), ms (1))ψa (r(2), ms (2))]
Spin singlet: [| tt) _ | tt)] /^2
Spin triplet: | tt) , | tt) , [| tt) + | tt)] /^2
2 Short answer problems
3 points each. Answer each of the following problems with a sentence, or a brief (1-line) calculation.
1. What is the energy density at a frequency of 5 .0 x 1012 Hz inside a blackbody of temperature 5000 K ?
2. A photon, initially of wavelength 5.2 x 10 — 12 m, is scattered off of an electron in the ground state of a Hydrogen atom, at an angle of θ = π/4. What is the wavelength of the scattered photon?
3. An atom in its excited state radiates energy within a time of 10 —6 s. What is the minimum uncertainty in the energy of this excited state?
4. An electron in the state ψ2 , 1 , 1 is placed in a magnetic field B = Bzˆ . Assuming that this magnetic field is so large that you can ignore any spin-orbit coupling effects, by how much can the electron’s energy change? Please give 2 possibilities.
5. Two electrons are in a square well potential. If this system is in its 2-particle ground state, what is their total spin, and how many states are there?
6. For the potential shown, for what range of energies do you expect solutions to the time-independent Schroedinger equation for which the energy is quantized?
7. An electron is in the third (n = 3) level of a Hydrogen atom. What are the possible values for j , its total angular momentum quantum number?
8. Which of the following wave-functions are eigenstates of the angular momentum operator z ? (i) eimφ , (ii) sin(mφ), (iii) cos(mφ)
9. Is the following wave-function an acceptable wave-function for fermions in a square well potential V (x) = 0 for _L/2 < x < L/2, with V (x) = o elsewhere? Do not worry about normalization.
ψ(x1 , x2 ) = ╱sin cos _ cos sin 、 (| tt) _ | tt))
10. Two electrons are in the first excited state of the Helium atom, and both have orbital angular momentum 0. What is the total spin angular momentum of the state(s) which have lowest energy if we ignore spin-orbit coupling, but include the effect of the Coulomb repulsion V (r1 , r2 ) = bewteen the 2 electrons?
3 Long Answer problems
18 points each. Please choose 4 of the following 5 questions. Show your work! If you turn in all 5 problems, only the first 4 will be graded.
1. An electron is known to be in the n = 3 shell of Hydrogen. In the following, consider only the Coulomb potential and the effect of spin-orbit coupling,
VSOC = S . L = (1.09 x 10 —4 eV A3 ) S.方2L
(a) How many different energies can our electron have?
(b) Given that
| (r2 dr) | (sin θdθ) | dφ ┌ ┐ =
what is the lowest possible energy that an electron in the n = 3 shell can have? How many states have this energy?
(c) If the electron relaxes from one of these lowest-energy states to a state in the n = 1 shell, what is the wavelength of the photon emitted?
(d) Are any of the lowest energy states you identified in (b) eigenstates of µz , the total magnetic dipole moment in the zˆ direction? Explain why or why not.
2. Consider the 1-dimensional potential
,.o x < 0
V (x) = .0 0 < x < L
.(V0 x 2 L
(a) Write a general expression for the wave-function ψ(x) that solves the time-independent Schroedinger equation in each of the two regions 0 < x < L, and x > L, for an energy 0 < E < V0 . Wher- ever possible, include expressions describing how to relate parameters such as k in your general solution to E and V0 .
(b) What condition should your wave function satisfy at x = 0? What is ψ(x) for x < 0? Using this, you should be able to write the wave function in (a) in terms of two undetermined constants.
(c) Write two equations that these two constants must satisfy. You do not have to solve these equations . Do the solutions represent bound states?
(d) Suppose that you choose L and V0 so that the two lowest- energy states are described by the spatial wave-functions ψ1 (x) (energy _0.8eV) and ψ2 (energy _0.2eV). If you put 2 electrons into this potential, what are all possible wave-functions for the first excited state of your 2-particle system?
3. A black body radiates at a temperature T = 3500 K.
(a) What is the most probable frequency for photons to be emitted at? Note that the equation ex (3 _ x) = 3
is solved (approximately) by x = 2.821.
(b) If we direct only the radiation at the frequency found in (a) onto a metal radiation at the frequency found in (a) onto a metal, what is the largest the work function of the metal can be for a non-vanishing photo-current to be observed? (If you are unable to answer (a), use a frequency of ν = 1.9 x 1014 Hz. This is not the correct answer for (a)!)
(c) Sodium has a work function w0 = 2.75 eV. If we direct all of the radiation emitted from our black-body onto some sodium, will we see any photo-current? If so, how will its intensity compare to the intensity of the photo-current from an object with the work function you calculated in (b)? You do not need to give a precise numerical value, but describe whether it will be roughly the same, (much) larger, or (much) smaller.
(d) If you reduce the black body’s temperature by a factor of 1/2, how will your answers in (b) and (c) change?
4. At time t = 0, an electron in a hydrogen atom occupies the quantum state
ψ(r, θ, φ) = N ╱3ψ200 _ 2ψ100 +^2ψ210 、
where N is a normalization factor. For this problem, you will find it helpful to note that
| d3 rψn(*),l,mlψn\ ,l\ ,m =
(a) Is this a solution to the time-independent Schroedinger equation for Hydrogen, in the absence
of spin-orbit coupling? Using your answer, determine whether (E) is independent of time. (b) If you measure 2 , what are the possible values you can find?
(c) What values can you find if you measure X ?
(d) Calculate N .
(e) Calculate (E2 ) _ (E) 2 (E here being the energy) for this state.
(f) Calculate (2 ) _ () 2 for this state.
5. A beam of atoms propagates along the yˆ direction, and passes through a Stern-Gerlach apparatus with field gradient in the zˆ direction. Each atom is in its ground state, and has total angular momentum 1. (You can think of this as implying that each atom has spin 1). Recall that
X = 方 00_1ì(、)
In the following, when describing the pattern emerging from a Stern-Gerlach apparatus, please specify the relative intensities of all bands.
(a) If the beam of atoms entering the device is initially in a random spin state, sketch what the output pattern looks like. Specifically, draw the pattern, and indicate the associated value of SX for each band observed, as well as the relative intensities of the bands.
(b) Suppose instead that the beam of atoms entering the device all are in the spin state
1 ╱ 0(1) 、
^2 ( _1ì
which is an eigenstate of the operator
x = 方 ╱、
with eigenvalue 0. If we input this state into the apparatus with field gradient along zˆ, how does your pattern in (a) change?
(c) Next, we add a second Stern-Gerlach apparatus, with field gradient in the direction. The atoms found to have Sz = 方 in aparatus (1) are directed into the second apparatus. What pattern is observed? Note that the other two eigenvectors of x are
(1, _^2, 1)T , (1, ^2, 1)T ,
with eigenvalues _方 and 方, respectively.
(d) If instead you re-focus all of the atoms exiting the zˆ apparatus into a single beam, and input this into the device, what do you observe? As in (b), assume the atoms enter the first device
in the state
1 ╱ 0(1) 、
^2 ( _1ì .
2023-05-06