Quantum Mechanics 1 | Week 7

Session: JAN-APR 2024

Course name: Quantum Mechanics I

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These are Quantum Mechanics 1 Week 7 Assignment Nptel Answers

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Q1.

Answer: b)

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Q2. The sum of two non-interacting angular momenta J₁ and J₂ is given by J = J₁ + J2. Then which of the following commutation relations are correct?
a) [Jr, Jy] =-ih.Jz
b) [Jy, Jz]=-ihJ
c) [J2, J²] = 0
d) [J2, J2] = 0
(Here J² is the total angular momentum and Jr, Jy, J₂ are the x, y, z components of the angular momentum operator respectively.)

Answer: c), d)

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Q3. Consider a wavefunction (r, 0, 0) = C1Wmhmr + C2n212m2 which is a combination of normal- ized stationary state Hydrogen-like wavefunctions nim. For to be normalized c₁ and c₂ must satisfy the condition |C1|²+ |C1|² = 1.
a) True
b) False

Answer: b) False

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Q4. At time t = 0, the wavefunction of the Hydrogen atom is
(r, 0) = 1 √10 (22/100+210 + √2211 + √3/21,-1)
where the subscripts are the values of the quantum numbers n, l, m. The expectation value of the energy of the system is
a) (E) = 1/10(2E1+ 4E2)
b) (E) = 1/10(2E1+ 6E2)
c) (E) = 1/10(4E1 – 6E2)
d) (E) = 1/10(4E₁ + 6E2)
where E1, E2 are the energies of the first and second excited states respectively.

Answer: d) (E) = 1/10(4E₁ + 6E2)

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Q5. The kets j, m) are the simultaneous eigenkets of total angular momentum operator J2 and its z component J₂ with eigenvalues j(j + 1)ħ² and mh respectively. Then J+j, m) and J_j, m) are also eigenkets of J2 with same eigenvalue.
a) True
b) False

Answer: a) True

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Q6. One real operator A satisfies the equation A2-5A +6 = 0. Is A an observable?
a) Yes
b) No

Answer: a) Yes

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Q7. Using the concept of parity, find out which of the following integrals are non-zero.
a) (2s|x|2py)
b) (2s|x2|2px)
c) (2px|x2|2px)
d) (2p|x|3d)
Here the functions in the integrals are Hydrogen-like wavefunctions.

Answer: a), c), d)

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Q8.

Answer: 0.6

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Q9. Consider a system of two non-interacting identical particles of mass m confined in a two-dimensional box of sides a. The energy of the system in the state 2,2(x1, y1) 1,4 (x2, y2) will be N²pi²h²/2ma², where N = ___________.

Answer: 5

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Q10.

Answer: d)

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