Quantum Mechanics Preparation
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1) 𝐿± = 𝐿𝑥 ± 𝑖𝐿𝑦 are raising and lowering angular momentum operators.
i) Prove that L±|l, m are also eigenfunctions of L2. What are its eigenvalues
corresponding to those eigenfunctions?
ii) What are the Harmonic conjugate operators of these raising and lowering
operators?
iii) Given |𝑙, 𝑚⟩ is eigenfunction of L2 . Show that it can be eigenfunction of
𝐿− under specific case. Find the 𝑙, 𝑚 values for the case.
iv) What is the parity of angular momentum operator. Justify your answer.
2) Consider a potential in one dimensional problem which confines a particle of mass m in
a small region (consider dimension a) to its ground state. At timet0, the potential suddenly
disappears.
i) Find the wavefunction (in terms of position and time) at time t > t0 in terms of
wavefunction at t = t0.
ii) Consider Gaussian wave packet for the wavefunction at t = t0. Give an expression for
the probability per unit time that the particle arrives at time t > t0 at an observer sitting
L distance away.
3) Consider two electrons are at orbital angular momentum quantum number states (one at
l=0, and other at l=1). Radial orbital wavefunctions of the states are R0(r) and R1(r). Also
consider, electrons are not interacting.
i) Estimate the possible total spin angular momentum (S), total orbital angular momentum (L)
and total angular momentum (J) of the atomic configuration. You may write the states in
multiplet forms, 2S+1LJ.
ii) Express the above multiplet states in terms radial wavefunctions, spin and orbital angular
momentums of individual electrons.
iii) Mention the state which has the minimum energy among the multiplet states?
Suppose you want to absorb light and excite from ground state to highest energy multiplet
state. What will be the polarization of light?
4) A spin-1/2 particle of mass m swings in a potential V = 1/2mω2r2 in 3 dimensions.
Consider a perturbation Hamiltonian 𝛼𝜎̂. 𝑟̂ with Pauli spin operator 𝜎̂, position operator 𝑟̂ and
real constant α.
i) Express the unperturbed ground and first excited state energies and their degeneracy factors.
ii) Compute the first (in terms of order of perturbation) non-zero perturbation correction of the
ground state energy.
5) i) A wave corresponding to a quantum particle is found to have a wavelength = 2Å and frequency
= 1GHz. Calculate the values of the energy, momentum and mass of the particle.
iii) The work function of Sodium is 2.5 eV. What is the maximum wavelength of light that will
cause photo emission?
6) Consider a hydrogen atom with Bohr radius ‘a’ and the ground state wavefunction
𝜓(𝑟, 𝜃, 𝜙) = ℵ exp(𝛼𝑟 𝛽 ), where ℵ, 𝛼, and 𝛽 are constant.
i) Derive ℵ, 𝛼, and 𝛽.
ii) What is the expectation value < r > of r?
iii) What are the expectation values of the kinetic energy and the potential energy of the
electron? How are the two related?
iv) Justify that a rotating electron around the nucleus of hydrogen atom does not
collapse or not collapse to nucleus
7) A nuclear–spin zero hydrogen atom is prepared in a magnetic field Bz (means atom has up and
down spin along z-direction). A uniformly increasing magnetic field along the x-direction is
switched on at t = 0 and after time t=T it becomes constant. Ignore higher power of (𝐵𝑥 /𝐵𝑧 ).
i) Find the potential along x-direction.
ii) Find the state of the electron at t = T.
iii) Find the time limit T, so that the system will be linear to Bx. [3+5+2=10]
8) Consider a quantum particle.
i) Prove that linear and orbital angular momentum are Hermitian.
ii) Explain the violation properties of angular momentum if the commutation relation is
[𝐿𝑥 , 𝐿𝑦 ] = ℏ𝐿𝑧
iii) Write the transformation operator if you take a particle ‘a’ distance away along x-
direction, then ‘b’ distance along y, and brought back to original position.
Find the phase change of the particle.
9) Drive the matrix representation of spin operators Sx, Sy and Sz of a spin-1 particle in the basis of
spin-state along z-direction.
10) Consider a spin-1/2 particle. The wavefunction |Ψ⟩ has following components: |Ψ± (𝑟⃗)⟩ =
⟨𝑟⃗, ±|Ψ⟩. |±⟩ indicates spin-up or down along z-axis.
1 1 1
Assume that |Ψ+ (𝑟⃗)⟩ = 𝑅(𝑟) [ 3 𝑌11 (𝜃, 𝜙)] and |Ψ− (𝑟⃗)⟩ = 𝑅(𝑟) [ 3 𝑌−1 (𝜃, 𝜙)].
√ √
i) Find the condition of R(r) so that |Ψ⟩ is normalized.
ii) Estimate ⟨𝐿𝑧 ⟩, ⟨𝑆𝑥 ⟩ and ⟨𝑆𝑧 ⟩.
iii) If the measurement 〈𝐿2 〉 with respect to state |Ψ⟩ is zero. Specify the state described by
the particle after this measurement.
