Quantum Mechanics 1 | Week 6

Quantum Mechanics Week 6 Answers

Course Link: Click Here

Q1. Consider a system whose Hamiltonian is given by H = (|0⟩⟨1| + |1⟩⟨0|)(2 + 2|1⟩⟨1|), where a is a real number having dimensions of energy and |1⟩, |2⟩ are normalized eigenstates of a Hermitian operator A that has no degenerate eigenvalues. Is H a projection operator?
a) Yes
b) No

Common data for Q2 & Q3

Consider a one-dimensional harmonic oscillator with Hamiltonian H = p² + mw²x². The position and momentum operators in the Heisenberg picture for this harmonic oscillator are respectively given as
XH(t) = X cos(wt) + P sin(wt) mw
PH(t) = P cos(wt) – mw x sin(wt).
Then the commutators

Q2. [PH(t₁), PH(t₂)] will be
a) iħmw sin[w(t₁-t₂)]
b) -ih/mw sin[w(t₁-t₂)]
c) -ihmw cos[w(t₁-t₂)]
d) -ih/mw cos[w(t₁-t₂)]

Q3. [XH(t₁), XH(t₂)] will be
a) -ihmw sin[w(t₁-t₂)]
b) -ih/mw sin[w(t₁-t₂)]
c) -ihmw cos[w(t₁-t₂)]
d) -ih/mw cos[w(t₁-t₂)]

Q4. Consider a system of two spin-half particles, in a state with total spin quantum number S = 0. If the spin Hamiltonian is H = AS₁·S₂, where A is a positive constant and S₁, S₂ are spin angular momentum operators, then the eigenvalue of the Hamiltonian will be
a) -3/2Ah²
b) 3/2Ah²
c) -3/4Ah²
d) 3/4Ah²

Q5.

Answer: a,d

Q6. For the case, n = 2, l = 1, m₁ = 0, the value of r at which the radial probability density of the hydrogen atom reaches its maximum is _ a₀.
Answer: 4

Q7. The Hamiltonian of a system is given by H(t) = T + V(R(t)). The Heisenberg equation of motion for the position operator R(t) in the Heisenberg picture for this system can be written as
a) dR/dt = P²/2m
b) dR/dt = P/m
c) dR/dt = P/2m
d) dR/dt = P/2iħm

Q8. The first excited state of Hydrogen atom is __ fold degenerate when we take into account the spin of the electron.
Answer: 4

Q9.

Answer: c

Q10. The expectation value of a physical observable in Schrodinger’s representation (S) is the same as its expectation value in Heisenberg’s representation (H).
a) True
b) False