Quantum Mechanics 1 | Week 5

Quantum Mechanics Week 5 Answers

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Q1. Consider the states |4⟩ = 3|1⟩ + 7i|2⟩ and |x⟩ = -|1⟩ + 2i|2⟩, where |1⟩ and |2⟩ are orthonormal. The states satisfy the triangle inequality.
a) True
b) False

Q2. Consider two operators A and B which are Hermitian and commute. Then the operator (A+B)/√(A² + B²) is unitary.
a) True
b) False

Q3. Consider a one-dimensional particle with the wavefunction ψ(x,t) = sin(x/a) exp(-iwt)which is confined within the region 0 ≤ x ≤ a. The probability of finding the particle in the interval a/4 < x < 3a/4 is __. (Write up to two decimal places.)

Answer: 0.82

Q4. A system is initially in the state |ψ⟩ = [√2|1⟩ + √3|2⟩ + √3|3⟩ + √4|4⟩]/√7, where |n⟩ are the eigenstates of the system’s Hamiltonian such that H|n⟩ = n²E₀|n⟩. If energy E₂ is measured, its value and the probability will be respectively,
a) 480, 3/7
b) 480, √3/√7
c) E₀, 3/7
d) E₀, √6/√7
(Here E₀ is the ground state energy of the system.)

Q5. If ⟨l, m|[Lz, A]|l, m⟩ = iħ for any observable A, then the expectation value of 2p is
a) -i
b) 1
c) 0
d) -1
Q6. Consider the one-dimensional motion for a free particle and three operators, the Hamiltonian H, the momentum operator p and the parity operator P. Which of the following are commuting?
a) [H, p²]
b) [HP]
c) [P, p²]
d) [P,p]

Q7. The initial wavefunction of a particle in the harmonic oscillator potential (mw²²) can be written as
(х, 0) = A[340(x) + 441(x)],
where A is the normalization constant, 40 and 1 represent the ground state and first excited states of the harmonic oscillator respectively and are orthogonal. Then (x, t)|2 can be written as b + c + d4041 cos(wt)], where
i) a = _______(Answer should be an integer)
ii) b = _______(Answer should be an integer)
iii) c = _______(Answer should be an integer)
iv) d = _______(Answer should be an integer)

Answer: 25,9,16,24 

Q8. The Hamiltonian operator for a two-state system is given by H = a[2|1⟩⟨1| – 13|2⟩⟨2|], where a is a number with dimensions of energy. The corresponding eigenvalues and eigenvectors will be
a) a(1±√2); ((-√2|1⟩ + |2⟩), (√2|1⟩ + |2⟩))
b) ±a; ((|1⟩ + |2⟩), (|1⟩ – |2⟩))
c) a, -3a; ((3|1⟩⟨1| + 2|2⟩⟨2|), (|1⟩ + |2⟩), (|1⟩ + |2⟩))
d) -a, 3a; ((-3|1⟩⟨1| + 2|2⟩⟨2|), (|1⟩ + |2⟩))

Q9. Suppose a particle is in the state |ψ⟩ = (3√2i|1⟩ + √7/3|2⟩), where {|1⟩, |2⟩, |3⟩} form an orthonormal basis. For an observable B = a|1⟩⟨1| + b/3|3⟩⟨3|, the probability of measuring b in such a state is
(a) 0
(b) 3/10
(c) 8/25
(d) 7/20

Q10. In a system described by three orthonormal basis {|1⟩, |2⟩, |3⟩}, the operator form of an observable is A = 7|1⟩⟨1| + 9√2|2⟩⟨2| -5/3|3⟩⟨3|. The possible measurable values of observable A will be
a) -7, -9, 5
b) 7, 9, -5
c) 1/7, 1/9, -1/5
d) -1/7, -1/9, 1/5