# Quantum Mechanics 1 | Week 10

Session: JAN-APR 2024

Course name: Quantum Mechanics I

#### Q1. When a state vector transforms into another state vector by a unitary transformation, an operator A transforms as A’. If A is Hermitian, then A’ is anti-Hermitian.a) Trueb) False

Q2. The wavefunction of an electron in a Hydrogen-like atom is (r) = Ce-r/a, where a = ao/Z and an is the Bohr radius. The normalization constant C is
a) 1/√4πa3
b) 1/√2πa3
c) 1/√πa3
d) 1/√a3

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Q3. The eigenstate of Jx with eigenvalue ħ/2 will be
a) 1/√8(-√3 -1 1 √3)
b) 1/√8(1 √3 √3 1)
c) 1/√2(-√3 -1 1 √3)
d)1/√2(1 √3 -1 -√3)

Q4. Consider an operator = i.Jx.Jy. The expectation value of this operator for j = m = 1/2 is
a) 1/2h2
b) 1/4h2
c) -1/2h2
d) -1/4h2

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Q5. If (r) and (r) are the eigenfunctions of the parity operator belonging to even and odd eigenstates respectively, then they are orthogonal.
a) True
b) False

Q6.

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Q7. Classically, the rotational energy of a molecule is given as 2π²Iv², where I is the moment of inertia of the molecule and v is the number of revolutions per second. Moving to Quantum Mechanics, if such a rigid diatomic molecule is in the state l = 2, then the number of revolutions per second will be
a) √6h/πI
b) 6h/πI
c) 6h/2πI
d) √6h/2πI

Q8. For a harmonic oscillator, the Hamiltonian in dimensionless units (m = h = w = 1) is
where the creation and annihilation operators are defined as,
The energy eigenfunction of a state is
The quantum number related to this state is

a) n = 2
b) n = 3
c) n = 4
d) n = 5

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Q9. If x’ and p’x are the coordinate and momentum after time reversal, then the fundamental commu- tation [x’, p] is-ih.
a) True
b) False

Q10. The value of (1.0) [J², J.] [1,0) is
a) 0
b) h
c) -h
d) -h²