Quantum Mechanics 1 | Week 4

Quantum Mechanics Week 4 Answers

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Q1. If operator A is Hermitian, then the operator B = iA is Hermitian.
a) True
b) False

Q2. Which of the following statements about parity operator are correct?
a) Parity operator P is anti-Hermitian.
b) P is unitary.
c) P² is equal to unity operator Î.
d) The eigenvalues of P are +1 or -1.

Q3. Consider the states |ψ⟩ = 9i|1⟩ + 20|2⟩ and |x⟩ = -|1⟩ + 12|2⟩, where the vectors |1⟩ and |2⟩ form a complete and orthonormal basis. Then Tr(|ψ⟩⟨x|) is a/√b, where
i) a = (Answer should be an integer)
ii) b = (Answer should be an integer)

Answer: 7, 2

Q4. By evaluating the commutator [A, [B, C]D] we get
a) CBDA – BCDA + ABCD – ACBD
b) ABCD – CDBA + BACD – DCBA
c) CBDA + BCDA – ABCD + ACBD
d) ABCD – BCDA + DCBA – DABC
Here A, B, C, D are linear operators.

Q5. Given that, |V⟩ = (4+3i)|V₁⟩ – 4 exp(-3/2)|V₂⟩, where |V₁⟩ and |V₂⟩ are an orthonormal pair of kets. Then the norm of |V⟩ is √a, where, a = _. (Answer should be an integer.)

Answer: 41

Q6. Consider a three-dimensional vector space spanned by an orthonormal basis |1⟩, |2⟩, |3⟩. Kets |a⟩ and |β⟩ are given by
|a⟩ = |1⟩ – 2|2⟩ – 3|3⟩, |β⟩ = |1⟩ + 2/3|3⟩
Then the inner product ⟨a|β⟩ = a + bi, where
i) a = (Answer should be an integer)
ii) b = (Answer should be an integer)

Answer: 1, 2

Q7. Consider an unnormalized state which is given in terms of three orthonormal vectors as follows.
where |n⟩ are eigenstates of an operator B such that B|n⟩ = n²|n⟩ with n = 1, 2, 3. Then the expectation value of B for the state would be a/b where
i) a = (Answer should be an integer)
ii) b = (Answer should be an integer)

Answer: 11, 4

Q8. Consider a pair of orthogonal kets |V₁⟩ and |V₂⟩. The dual of the ket |V⟩ = (5+3i)|V₁⟩ + 3exp(-13/2)|V₂⟩, will be
a) (5+3i)|V₁⟩ + 3exp(3/2)|V₂⟩
b) (5+3i)(|V₁⟩ + 3exp(3/2)|V₂⟩)
c) (5+3i)(|V₁⟩ + 3exp(-13/2)|V₂⟩
d) (5+3i)(|V₁⟩ – 3exp(23/2)|V₂⟩

Q9. Consider an operator B = |1⟩⟨2| where |1⟩ and |2⟩ are orthonormal. Then the correct statement about B² and BB is
a) B² is |1⟩⟨1| and BB is 0.
b) B² is |1⟩⟨2| and BB is a projection operator.
c) B² is 0 and BB is a projection operator.
d) B² is a projection operator and BB is 0.

Q10. The parity of both the position operator R and momentum operator P are odd.
a) True
b) False