Download Quantum Mechanics Problems and Solutions and more Exams Quantum Mechanics in PDF only on Docsity!
August 2023
QUANTUM MECHANICS TEST BANK
- Kinematics of Quanta [500 level] An electron moving along the positive z-axis with the huge γ factor of γe = 10^6 , or the energy Ee = 511 GeV, hits a very soft CMB photon with an energy Eγ = 10−^3 eV moving in the opposite direction (along the negative z-axis). After the collision the direction of the three-momentum of the incoming photon is reversed, while the electron continues to move along the positive z-xis. (a) Derive the appropriate inverse Compton scattering formula for the energy E γ′ of the scattered photon and compute its numerical value. Use the leading term in the asymptotic expansion in the γ-factor of the incoming high energy electron. (b) Compute the approximate numerical value of the energy E γ′ of the scattered photon when the initial photon energy is Eγ = 1 eV. (c) Can you use the same approximate formulas when 4 (^) (EmceE 2 γ) 2 is larger than 1? Find the appropriate approximate formula in this case.
- Unknown Potential [500 level] A one-dimensional system has the ground state wave function ψ(x) = N e−x (^4) /a 4 , where N is a normalization constant. (a) Write an integral expression for the normalization constant N. (b) Find the potential if the ground state energy is E 0.
- One-Dimensional Potential [500 level] A particle of mass m moves in one dimension, subject to a δ-function potential V (x) = −Kδ(x). (a) Find the matching condition for the wave function across the δ-function at x = 0. (b) Show that this potential has exactly one bound state. (c) Find the bound state energy.
- Tunneling [700 level] Consider a particle in a one-dimensional double-well potential
V (x) = V 0
x a
x a
(a) Show that the eigenstates of the Hamiltonian are also eigenstates of the parity operator. Sketch the potential and the two lowest energy eigenstates.
(b) At time t = 0, the particle is localized on one side of the double well, in the state ψ(x, t = 0) =
[ψ+(x) + ψ−(x)],
where the ψ± are the eigenstates with energies E+ < E−. Write down the time-dependent wave function ψ(x, t). What is the period of oscillation of the particle? (c) Using the WKB approximation, estimate the energy difference E− − E+.
- One-Dimensional Scattering [700 level] A particle of mass m moves in a potential V (x) = α[δ(x) + δ(x − a)]. (a) Write down the matching conditions for the wave function across the δ- function singularities in the potential. (b) Determine the time-independent wave function for a particle with wave number k. (c) At what energies E will the particle pass through the potential without any reflection?
- Time-Dependent Potential [700 level] At times t < 0, a particle is in the ground state of the one-dimensional potential
V (x) =
0 , |x| < a V 0 |x| ≥ a
where V 0 is so large (≫ ¯h^2 / 2 ma^2 ) as to be effectively infinite. (a) Find this initial wave function. (b) At t = 0, the potential instantly changes to V = 0 everywhere. Find the wave function for times t > 0.
- Baker-Campbell-Hausdorff Lemma [500 level] (a) Demonstrate the Baker-Campbell-Hausdorff Lemma: If [A, [A, B]] = [B, [A, B]] = 0, then eAeB^ = eA+B^ e (^12) [A,B] . (b) Show that for the operators x and p,
eiαxeiβp^ = eiϕeiβpeiαx
for some phase ϕ, and find ϕ. (c) What are the conditions for eiαx^ and eiβp^ to commute?
- Two-State System [500 level] Consider a system that has only two linearly independent states,
and | 2 ⟩ =
(b) Define another operator g+(η) = U (η)(a + a†)U (η)†. Show that g+ satisfies the differential equation dg+ dη
= 2g+(η).
(c) Find dg−/dη for the operator g−(η) = U (η)(a − a†)U (η)†. (d) Use the result from parts (b) and (c) to show that
U (a ± a†)U †^ = e±^2 η(a ± a†).
- Commutators [500 level] Calculate the following commutators involving the angular momentum opera- tors: (a) [Lx, x] (b) [Lx, y] (c) [Ly, z] (d) [Lz , x] (e) [Ly, zx]
- Commutators [500 level] Calculate the following commutators involving the angular momentum opera- tors: (a) [Lx, px] (b) [Ly, px] (c) [Lz , py] (d) [Lx, pz ] (e) [Ly, pz ]
- Angular Momentum [500 level] Evaluate the action of the angular momentum operator in the following ex- pressions, involving r =
p x^2 + y^2 + z^2 and the azimuthal angle ϕ: (a) Lz kr (b) Lz sin kr (c) Lz eiϕ
- Angular Momentum [500 level] Consider a particle in a state with total angular momentum ℓ(ℓ + 1)¯h^2 and z projection m¯h. (a) Using L+ and L−, show that ⟨Lx⟩ = ⟨Ly⟩ = 0. (b) Using L^2 , show that ⟨L^2 x⟩ = ⟨L^2 y⟩ = 12 [ℓ(ℓ + 1) − m^2 ]¯h^2.
- Addition of Angular Momentum [700 level] What are the possible values of the total orbital angular momentum ℓ for the
following combinations of electrons? (a) Four p electrons (b) Three p and one f electron
- Molecular Hamiltonian [500 level] A rigid diatomic molecule, free to rotate around its center of mass, is given by
H =
L^2 − L^2 z 2 I 1
L^2 z 2 I 3
(a) What are the eigenfunctions (expressed in terms of the standard basis of Yℓm spherical harmonics) and corresponding eigenenergies for this system? (b) What are the degeneracies of the energy spectrum, and how are they related to the symmetry of this system? (c) The momenta of the inertia of the molecule obey this relation,
2 I 1
I 3 − I 1
I 1 I 3
8 ma^20
for electron mass m and Bohr radius a 0. Starting in the ground state, what energy is necessary to excite the system into the second excited state? Express the answer in terms of the Rydberg energy.
- Operator Algebra [500 level] Consider the vector operator Θ = L × r − ih¯r. (a) Show that Θ may also be written Θ = i¯hr − L × r. (b) Show that this operator is Hermitian. (c) Show that [L^2 , r] = − 2 ih¯Θ.
- Symmetries [700 level] Consider the following matrix elements. Some of them can be shown to be zero. State which ones are zero and give a brief reason for each answer. Each state has the form ϕ(j)^ m, where the superscript and subscript denote the total angular momentum and its z-projection quantum numbers. (a) ⟨ψ(3) 2 |f (r)L · S|ϕ(2) 2 ⟩ (b) ⟨ψ(3) 2 |J+|ϕ(2) 1 ⟩ (c) ⟨ψ(3) 2 |μx + iμy|ϕ(2) 1 ⟩ (d) ⟨ψ(3) 2 |V (r)|ϕ(3) 2 ⟩ (e) ⟨ψ(3) 1 |Qzz |ϕ(1) 0 ⟩
- Time Operator [500 level] Suppose that there is a time operator T canonically conjugate to the Hamil- tonian, so that [T, H] = ih¯. (a) Consider the unitary operator U = eiαT^. Determine the action of this
finding the spins up and down along the x-axis? (b) If the x-axis spin up beam is sent through a y-axis Stern-Gerlach apparatus, what are the probabilities for the particles emerging to be spin up and spin down along y? (c) If, instead, the two beams from the x-axis Stern-Gerlach apparatus are recombined without being measured, and the recombined beam is sent through the y-axis apparatus, what are the probabilities of the particles coming out being spin up or spin down along y?
- Spin Sum [500 level] Consider a quantum system of spin 12. The spin operator is S = Sxˆi+Syˆj+Sz ˆk. (a) What are the eigenvalues and eigenvectors of the operator Sx + Sy? (b) Suppose a measurement of Sx + Sy is made, and the system is found to be in the eigenstate corresponding to the larger eigenvalue. What is the proba- bility that a subsequent measurement of Sz yields ¯h 2? (c) Alternatively, starting from the same Sx + Sy eigenstate, what is the prob- ability that a measurement of Sy yields ¯h 2?
- Total Angular Momentum [700 level] A 3 P 0 atomic state hs spin and orbital angular momenta s = ℓ = 1 but total angular momentum j = 0. (a) Although there are nine basis states |mL⟩ ⊗ |mS ⟩, show by application of Jz = Sz + Lz that only three of them can possibly be part of the 3 P 0 state. (b) By similar application of Jx and Jy, determine the expansion of the 3 P 0 state in the |mL⟩ ⊗ |mS ⟩ basis.
- Variational Principle [500 level points] Consider a particle moving in the one-dimensional potential V (x) = K|x|. (a) Find an upper limit on the ground state energy using a variational wave function ψ 1 (x) = C 1 exp(−|x|/a), where C 1 is the appropriate normalizaiton factor. (b) Find another upper limit using a different wave function ψ 2 (x) = C 2 exp(−x^2 /b^2 ). (c) Which function gives a tighter bound on the ground state energy?
- Variational Principle [700 level] Consider a system described by the Hamiltonian H. Let ψn (n = 0, 1 , 2 ,.. .) be the normalized eigenstates, Hψn = Enψn, ⟨ψn|ψn⟩ = 1. Let a normalized wave function |ϕ⟩ be expanded as
|ϕ⟩ =
X^ ∞
n=
cn|ϕn⟩.
(a) Find the condition on the cn coefficients imposed by the normalization of |ϕ⟩.
(b) With the help of the expansion, prove ⟨ϕ|H|ϕ⟩ ≥ E 0 for any |ϕ⟩. (c) Suppose we have calculated ⟨ϕ|H|ϕ⟩ for various choices of |ϕ⟩ and obtained ⟨H⟩min as the minimum value of ⟨ϕ|H|ϕ⟩. Since ⟨H⟩min ≥ E 0 , we may get a rather reliable estimate of the ground state energy as E 0 ≈ ⟨H⟩min. Now, consider the harmonic oscillator Hamiltonian
H =
¯h^2 2 m
d^2 dx^2
mω^2 x^2
and a family of normalized trial wave functions ϕβ (x) = N exp(−βx^2 ). Find the normalization constant N. (d) Calculate f (β) = ⟨ϕβ |H|ϕβ ⟩. (e) Find an estimate of he ground state energy E 0 by minimizing f (β). You may find the following integrals useful in your calculations: Z (^) ∞
−∞
dy exp(−γy^2 ) =
r π γ Z (^) ∞
−∞
dy y^2 exp(−γy^2 ) =
r π γ^3
- Identical Particles [500 level] Three identical bosons with spin s = 1 are placed into the same orbital state ϕ(r). (a) Count the number of possible states of the three bosons. (b) Among the states, which ones have total z spin projection mS = 0? (c) What are the possible values of the total spin of the three-boson system?
- Identical Particles [700 level] The normalization for a two-particle wave function Ψ(x 1 , x 2 ) is given by Z dx 1
Z
dx 2 |Ψ(x 1 , x 2 )|^2 = 1.
Let ψα(x) and ψβ (x) be normalized one-particle wave functions, Z dx |ψα(x)|^2 =
Z
dx |ψβ (x)|^2 = 1.
We allow for the possibility that ψα and ψβ are not orthogonal. (a) If the particles, 1 and 2, are distinguishable, what is the expression for the normalized wave function Ψ(x 1 , x 2 ) for the two-particle wave function? (b) Find the probability density for observing particle 1 at x 1 = a, regardless of the position of the distinguishable particle 2.
- Atomic Units [500 level] In atomic physics, it is natural to use units that match the scales in atoms. Mass is measured in units of the electron mass me. Energy is measured in the Hartree unit, mee^4 /h¯^2 (in Gaussian units); this is the magnitude scale of the kinetic and potential energies in a hydrogen atom. Length is measured in Bohr radii, a 0 = ¯h^2 /mee^2. Angular momentum is measured in units of ¯h. Magnetic moments are in units of Bohr magnetons, μB = e¯h/mec. This problem concerns the appropriate atomic units of electric and magnetic fields. All dimensionless factors of order unity may be neglected. (a) The natural unit of E is the field strength at a distance a 0 from a hydrogen nucleus. Express this E 0 in terms of fundamental constants. (b) There are two possible ways of formulating a unit of B. One possibility is the magnetic field at the nucleus from an electron in the first Bohr orbit. Find this field BN. (c) There is also the field that splits the spin-up and spin-down states of a particle with magnetic moment μB by 1 Hartree. Find this field BH. (d) Find the relationships between these two magnetic field units and E 0 , in terms of the dimensionless fine structure constant α = e^2 /¯hc. (e) Which of the units for B is related to the natural scale of magnetic energy shifts in atoms?
- Hydrogen Atom [500 level] The ground state wave function for an electron in a hydrogen atom has the form ψ(r) = N e−r/a^0 , where N is a normalization and constant and a 0 = ¯h^2 /me^2. (a) Find the average value of the electron kinetic energy ⟨T ⟩. (b) Find the average value of the potential energy ⟨V ⟩.
- Variational Principle in a Helium-Like Atom [700 level] A helium-like ion with two electrons has Hamiltonian
H =
p^21 2 m
p^22 2 m
Ze^2 r 1
Ze^2 r 2
e^2 |r 1 − r 2 |
Take as an ansatz for the ground state wave function
ψ(r 1 , r 2 ) =
Z^3 eff πa^30
e−Zeff^ (r^1 +r^2 )/a^0 ,
where Zeff is an unknown “effective” nuclear charge. If the two electrons did not repel, this (with Zeff = Z) would be the exact wave function.
(a) Find the expectation value of the energy for the proposed state. You will find the integral (^) Z
d^3 r 1
Z
d^3 r 2
e−λ(r^1 +r^2 ) |r 1 − r 2 |
20 π^2 λ^5 useful. (b) Find the value of Zeff that minimizes the energy. (c) For this kind of position wave function, what must the spin state of the two electrons be?
- Stark Effect [500 level] An electric field E = E 0 kˆ is applied to a hydrogen atom. (a) Find the correct basis for the perturbation acting on the degenerate n = 2 states with wave functions
ψ 2 S (r) =
2 πa^30 /^2
r a 0
e−r/^2 a^0
ψ 2 P 0 (r) =
cos θ 4
2 πa^30 /^2
r a 0
e−r/^2 a^0.
(b) Find the first-order Stark energy shifts for the basis of states found in part (a). (c) What are the first-order energy shifts for the remaining
ψ 2 P+1 (r) = −
eiϕ^ sin θ 8
πa 3 / 2 0
r a 0
e−r/^2 a^0
ψ 2 P− 1 (r) =
e−iϕ^ sin θ 8
πa^30 /^2
r a 0
e−r/^2 a^0.
states?
- Spherical Potential [700 level] A particle of mass m is confined to the interior of a hollow sphere of radius R. (a) What are the ground state wave function of the particle and the corre- sponding energy? (b) Calculate the pressure the particle exerts on the sphere. The pressure may be defined as p = −∂E/∂V.
- Spherical Potential [700 level] A particle of mass m moves in an attractive spherical potential
V (r) =
−V 0 , r < a 0 , r ≥ a
experiences a perturbation V with matrix elements as shown:
V =
V 11 V 12
V 12 ∗ V 22
(a) Find the corrections to the energies up to second order in perturbation theory. (b) Find the exact expressions for the eigenenergies. (c) Show that when the exact expression is expanded to second order, it agrees with the result from part (a).
- Perturbation Theory [700 level] A system of two coupled harmonic oscillators has Hamiltonian
H = H 0 + H 1
H 0 =
p^21 2 m
mω 12 x^21 +
p^22 2 m
mω^22 x^22 H 1 = λx 1 x 2 ,
with ω 1 ̸= ω 2. (a) Find the energy of the perturbed number state |n 1 , n 2 ⟩ to first order in λ. Hint: Remember, xi =
q ¯h 2 miωi (ai^ +^ a
† i ). (b) Find the second-order energy shift for the |n 1 , n 2 ⟩ state. (c) What is the condition on λ for the validity of perturbation theory?
- Perturbation Theory [700 level points] Consider an electron in a one-dimensional harmonic oscillator potential 12 mω^2 x^2 , placed in an electric field E pointing in the x-direction. (a) Write the Hamiltonianp H in terms of raising and lowering operators a = mω 2¯h x^ +^ √ i 2 mω¯h p^ and^ a
† (^) = pmω 2¯h x^ −^ √ i 2 mω¯h p. (b) Calculate the first-order energy shift of the state |n⟩ due to E. (c) Calculate the second-order energy shift of |n⟩. (d) Find the exact energy levels of this system, and compare the results to the first- and second-order perturbative shifts.
- Perturbation Theory [700 level points] A particle of mass m moves in a one-dimensional square well potential with a δ-function in the center of the well,
V (x) =
∞, |x| > a λδ(x), |x| < a
(a) Find the energy eigenvalues for λ = 0. (b) Find the energy levels to first order in λ ̸= 0, using perturbation theory.
(c) Find a transcendental equation for the exact eigenvalues, and show that expanding this equation to first order in λ gives the same result as part (b).
- Time-Dependent Harmonic Oscillator [700 level points] A one-dimensional harmonic oscillator with Hamiltonian
H 0 =
p^2 2 m
kx^2 2
has energy eigenvalues E(0) n = (n + 1/2)¯hω 0 , where ω 0 =
p k/m. Suppose the spring constant is changed according to k → k′^ = k(1 + ϵ), with ϵ ≪ 1. (a) Find the exact new eigenvalues, and expand the result up to second order in ϵ. (b) Now calculate the perturbed energies, treating the Hamiltonian as H = H 0 + ϵV , where ϵV = ϵkx^2 /2 is a small perturbation. Work to first order in ϵ. (c) Calculate the perturbed energy for the ground state to second order in λ, and show that the result agrees with the result from part (a).
- Unstable Potential [500 level] Consider the Schr¨odinger equation with potential
V (r) = −
K
r^2
with a positive constant K. (a) Show that if ψ(r) is a solution of the time-independent Schr¨odinger equa- tion, so is the wave function ψ(cr) with coordinates rescaled by a c > 0, although the energies of the two eigenfunctions are different. (b) Show that if ψ(r) is a normalizable wavefunction, so is ψ(cr). (c) Show that this potential has no lowest-energy state. (This phenomenon is known as “falling to the center.”)
- Carbon Atom [500 level points] The electron configuration of a neutral carbon atom is 1s^22 s^22 p^2. (a) Taking the approximation in which L^2 , S^2 , J^2 , and Jz are good quantum numbers, what possible values can they take in the ground state manifold? (b) Which of the quantum numbers are still good with the full relativistic Hamiltonian? (c) What values can the true good quantum numbers take in the ground state?
- Spins [500 level points] The spin components of a beam of spin-^12 atoms prepared in an initial state |ψ 0 ⟩ are measured, and the following probabilities are obtained: P (Sz = +) = 12 , P (Sz = −) = 12 , P (Sx = +) = 34 , P (Sx = −) = 14 , P (Sy = +) = 0.067, and P (Sy = −) = 0.933. From the experimental data, determine the input state.
- Time-Dependent Observables [500 level points] (a) Show that the probability of a measurement of the energy is time in- dependent for a general state |ψ(t)⟩ =
P
n cn(t)|En⟩^ that evolves due to a time-independent Hamiltonian. (b) Show that probability of measurements for other observables are also in- dependent of time if those observables commute with the Hamiltonian.
- Two-State Hamiltonian [500 level points] Consider a Hamiltonian
H =
h¯ 2
ω 0 ω 1 ω 1 −ω 0
(a) Diagonalize this Hamiltonian. Find the eigenvalues and eigenvectors.
(b) Find time dependence of the initial state
- Magnetic Precession [500 level points] Consider a spin-^12 particle with a magnetic moment μ. At time t = 0, the state of the particle is |ψ(t = 0)⟩ = |+nˆ⟩—up in the ˆn direction, with ˆn = (ˆx + ˆy)/
- The system is allowed to evolve in a uniform magnetic field B = B 0 (ˆx +ˆz)/
- What is the probability that the particle will be measured to have spin up in the y-direction after a time t?
- Observable Operator [500 level points] (a) Consider a two-state system with a Hamiltonian
H =
E 1 0
0 E 2
Another physical observable A is described by the operator
A =
0 a a 0
where a is real and positive. Let the initial state of the system be |ψ(0)⟩ = |a 1 ⟩, where |a 1 ⟩ is the eigenstate corresponding to the larger of the two eigenvalues of of A. Find this eigenstate. (b) What is the frequency of oscillation of the expectation value of A?
- Three-State System [500 level points] Let the matrix representation of the three-state system be
H =
E 0 0 A
0 E 1 0
A 0 E 0
using the basis states | 1 ⟩, | 2 ⟩, and | 3 ⟩. (a) If the state of the system at time t = 0 is |ψ(0)⟩ = | 2 ⟩, what is the probability that the system is in the state | 2 ⟩ at time t? (b) If, instead, the state of the system at time t = 0 is |ψ(0)⟩ = | 3 ⟩, what is the probability that the system is in the state | 3 ⟩ at time t?
- Composite Spin State [500 level points] Consider a state of two s = 12 spins,
|ψa⟩ =
(|+z ⟩ 1 |−z ⟩ 2 + |−z ⟩ 1 |+z ⟩ 2 ).
(a) Transform each of the spins into the x basis and find the resulting two- particle spin wavefunction. (b) Transform each of the spins into the y basis and find the resulting two- particle spin wavefunction.
- Infinite Square Well [500 level points] Consider an infinite square well of width L, centered at x = 0. (a) Find the wavefunction for the n-th eigenstate. (b) Find the expectation values of x and p as functions of n. (c) Find the uncertainties ∆x and ∆p as functions of n.
- Square Well States [500 level points] Consider an infinite square well of width L, extending from x = 0 to L. Find the probability that a particle lies in the region 3L/ 4 < x < L for the three lowest eigenstates of the Hamiltonian.
- Square Well [500 level points] A particle at t = 0 is known to be in right half of an infinite square well of width L, with a probability density that is uniform in the right half of the well. (a) What is the initial wavefunction of the particle? (b) Calculate the expectation value of the energy. (c) Find that the probability that the energy takes on its n-the eigenvalue.
- Half-Infinite Square Well [500 level points] Consider a potential
V (x) =
∞, x < 0 0 0 < x < L V 0 x > L
(a) Find a transcendental equation for the energy of the ground state of the system. (b) Find the ground state energy, to the lowest nonvanishing order in 1/V 0.
(b) Evaluate the resulting integral. (c) Find the position uncertainty ∆x(t) as a function of t.
- Gaussian Wavefunctions [500 level points] (a) Show that a wave packet that is Gaussian in position space is also Gaussian in momentum space. (b) Calculate ∆x∆p for a Gaussian packet. (c) Explain why, for a free particle, ∆x depends on time, but ∆p does not.
- Sinusiodal Wavefunction [500 level points] Consider a particle whose wavefunction is ψ(x) = A sin(p 0 x/h¯). (If you wish, you make take the wave function to be defined on a space of length L, with periodic boundary conditions.) (a) Is this an eigenstate of momentum? Find the expectation value of p. (b) Calculate the uncertainty ∆p. What are the possible results of a measure- ment of the momentum?
- Uncertainty Estimate [500 level points] Use the uncertainty principle to estimate the ground-state energy of a particle of mass m bound in the harmonic oscillator potential V (x) = 12 kx^2. How does this compare to the true ground state energy?
- Uncertainty Estimate [500 level points] Use the uncertainty principle to estimate the ground-state energy of a particle of mass m bound in the potential V (x) = a|x|.
- Uncertainty Estimate [500 level points] Use the uncertainty principle to estimate the ground-state energy of a particle of mass m bound in the potential V (x) = bx^4.
- Position Uncertainty [500 level points] Calculate the position uncertainty for a particle bound in an infinite square well of width L if (a) the particle is in the ground state; (b) the wave function is uniform across the well.
- Beam Wavefunction [700 level points] A beam of particles is described by the wavefunction
ψ(x) =
Aeip^0 x/¯h(b − |x|), |x| < b 0 , |x| > b
(a) Normalize the wavefunction, using an appropriate convention for contin- uum states.
(b) Sketch the wavefunction. (c) Calculate and sketch a plot of the momentum probability distribution.
- Angular Momentum [500 level points] Starting with the definition of the angular momentum, show that the orbital angular momentum operators in spherical coordinates are
Lx = i¯h
sin ϕ
∂θ
∂ϕ
Ly = i¯h
− cos ϕ
∂θ
∂ϕ
Lz = i¯h
∂ϕ
- Wave Function on a Circle [500 level points] Consider the normalized state |ψ⟩ for a particle of mass μ constrained to move ona circle of radius r 0 , given by
|ψ⟩ =
N
2 + cos(3ϕ)
(a) Find the normalization constant N. (b) Sketch the wavefunction. (c) What is the expectation value of Lz in this state?
- Angular Wave Function [500 level points] Consider the normalized state of a particle of mass m on a sphere given by
|ψ⟩ =
i √ 6
in the |ℓm⟩ angular momentum basis. (a) What are the probabilities that a measurement of Lz will yield 2¯h, −¯h, or 0¯h? (b) What is the expectation value of Lz? (c) What is the expectation value of L^2? (d) What are the Hamiltonian and the expectation value of the energy in this state? (e) What is the expectation value of Ly?
- Hydrogenic Wavefunction [500 level points] (a) Calculate the normalization constant N , for the radial hydrogenic wave- function R 10 (r) = N e−Zr/a^0. (b) Calculate the expectation value for ⟨r⟩.
