Quantum Mechanics 1 | Week 3

Quantum Mechanics Week 3 Answers

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Q1. The commutator of two Hermitian operators is Hermitian.
a) True
b) False

Q2. Which of the following operators are Hermitian?
a) d/dx
b) i*d/dx
c) ed/dx
d) eid/dx

Q3. Consider two operators A and B defined by, Αψ(x) = ψ(x) +x and Bψ(x) = (dψ/dx)+2ψ(x). Which of the following statements about their linearity is true?
a) A is linear, B is not linear
b) B is linear, A is not linear
c) both A and B are linear
d) both A and B are not linear

Q4. Which of the following statements about projection operators are true?
a) An operator P is said to be a projection operator if it is Hermitian and equal to its own square.
b) The operator |ψ><ψ| is a projection operator only when |ψ> is normalized.
c) Two projection operators are said to be orthogonal if their product is 0.
d) The product of any two projection operators is always a projection operator.

Q5. Consider a ket |ψ> = (5i 2 -i) which is not normalized. It can be normalized by multiplying it with a factor of 1/√a, where a = ______.

Answer: 30

Q6. Consider a ket |ψ> = (-1+i 3 2+3i) and bra <ⲫ|= (6 -i 5). The value of <ψ|ⲫ) will be a-bi.
Answer: 4, 18

Q7. The Hermitian adjoint of f(A) = (1 – 2iA +3A)(1 + iA – 24) is
a) (1-2iA†+ 3A†)(1 +2iA† – 2¹)
b) (1-iA† – 2A†)(1 +2iA† + 3¹)
c) (1 + 2iA – 3A)(1 -iA† + 2¹)
c) (1 – iA + 2A)(1 – 2iA† – 3¹)

Q8. Evaluate the commutator [x, [x, H]], where H is the Hamiltonian operator.
a) ħ²/m
b) -ħ²/m
c) ħ²/2m
d) -ħ²/2m

Q9. Consider the states |ψ> = 3i|ⲫ1> – 7i|ⲫ2> and |x> = -|ⲫ1> + 2i|ⲫ2>, where |ⲫ1> and |ⲫ2> are orthonormal. Then <ψ+ x|ψ + x> is _________________.

Answer: 35

Q10. Consider the states |ψ1> = 2i|ⲫ1> + |ⲫ2> + a|ⲫ3> + 2|ⲫ4> and |ψ2> = 2|ⲫ1> – 3|ⲫ2> + 3|ⲫ3> – |ⲫ4>, where |ⲫ1>, |ⲫ2>, |ⲫ3>, |ⲫ4> are orthonormal kets and a is a constant. The value of a so that |ψ1> and |ψ2> are orthogonal is
a) (7i-2)/3
b) (-7i+2)/3
c) (7i+2)/-3
d) (-6i-2)/3