# Quantum Mechanics 1 | Week 6

Session: JAN-APR 2024

Course name: Quantum Mechanics I

#### Q1. Consider a system whose Hamiltonian is given by H = (01) (2+2)(011), where a is real number having dimensions of energy and 1), 2) are normalized eigenstates of a Hermitian operator A that has no degenerte eigenvalues. Is H a projection operator?a) Yesb) 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(1), XH (2)] 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)]

Answer: b) -ih/mw sin[w(t1 – t2)]

These are Quantum Mechanics 1 Week 6 Assignment Nptel Answers

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.

Q6. For the case, n = 2,1 = 1, m₁ = 0, the value of r at which the radial probability density of the hydrogen atom reaches its maximum is ___________ a0. (Answer should be an integer)

These are Quantum Mechanics 1 Week 6 Assignment Nptel Answers

Q7. The Hamiltonian of a system is given by H(t) = + V(R(t)). The Heisenberg equation of 2m 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/2ihm

Q8. The first excited state of Hydrogen atom is ________ fold degenerate when we take into account the spin of the electron. (Answer should be an integer)

These are Quantum Mechanics 1 Week 6 Assignment Nptel Answers

Q9.

These are Quantum Mechanics 1 Week 6 Assignment Nptel Answers

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