MCQs On General Formalism in Quantum Mechanics with Answers

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MCQs On General Formalism in Quantum Mechanics with Answers

Q1. The ground state (apart from normalization) of a particle of unit mass moving in a one-dimensional potential V(x) is exp(-x2/2) cosh (√2x). The potential V(x), in suitable units, so that ℏ = 1, is (up to an additive constant).

(a) x2/2
(b) x2/2 – √2 x tanh(√2x)
(c) x2/2 – √2 x tan(√2x)
(d) x2/2 – √2 x coth(√2x)

Q2. The wave function of a free particle in one dimension is given by ψ(x) = Asin x + Bsin 3x. Then ψ(x) is an eigenstate of

(a) the position operator
(b) the Hamiltonian
(c) the momentum operator
(d) the parity operator

Q3. The hermitian conjugate of the operator (-/∂x) is

(a) /∂x
(b) –/∂x
(C) i /∂x
(d) -i /∂x

Q4. If the expectation value of the momentum is <P> for the wavefunction ψ(x), then the expectation value of momentum for the wavefunction eikx/ℏ ψ(x) is

(a) k
(b) <P> – k
(c) <P> + k
(d) <P>

Q5.  The Hamiltonian operator for a two-state system is given by

H = a(|1><1| – |2><2| + |1><2| + |2><1|)

where ‘a’ is a positive number with the dimension of energy. The energy eigenstates corresponding to the larger and smaller eigenvalues respectively are

(a) |1> – (√2 + 1)|2>, |1> + (√2 – 1)|2>
(b) |1> + (√2 – 1)|2>, |1> – (√2 + 1)|2>
(c) |1> + (√2 – 1)|2>, (√2 + 1)|1> – |2>
(d) |1> – (√2 – 1)|2>, (√2 + 1)|1> + 2>

Q6. The Operators A and B share all the eigenstates. Then the least possible value of the product of uncertainties ΔAΔB is

(a) ℏ
(b) 0
(c) /2
(d) Determinant (AB)

Q7. A particle of mass m moves in a 1-dimensional potential V(x), which vanishes at infinity. The exact ground state eigenfunction is ψ(x) = A sech (λx) where A and λ are constant. The ground state energy eigenvalue of this system is,

(a) E = (ℏ2λ2)/m
(b) E = -(ℏ2λ2)/m
(c) E = -(ℏ2λ2)/m
(d) E = (ℏ2λ2)/m

Q8. Given that ψ1 and ψ2 are eigenstates of a Hamiltonian with eigenvalues E1 and E2 respectively, what is the energy uncertainty in the state (ψ1 + ψ2)?

(a) -√(E1E2)
(b) 1/2 |E1 – E2|
(c) 1/2 |E1 + E2|
(d) 1/√2 |E2 – E1|

Q9. If a Hamiltonian H is given as

H = |0><0| – |1><1| + i|0><1| – |1><0|,

where |0> and |1> are orthonormal states, the eigenvalues of H are

(a) ± 1
(b) ± i
(c) ± √2
(d) ± i√2

Q10. The adjoint of a differential operator d/dx acting on a wavefunction ψ(x) for a quantum mechanical system is:

(a) d/dx
(b) – iℏ d/dx
(c) – d/dx
(d) iℏ d/dx

Q11. If ψ(x) is an infinitely differentiable function, then Dψ(x), where the operator D = exp(ax d/dx) is

(a) ψ(x + a)
(b) ψ(aea + x)
(c) ψ(ea x)
(d) ea ψ(x)

Q12. The wave function of a quantum mechanical particle is given by

ψ(x) = 3/5 φ1(x) + 4/5 φ2(x)

where φ1(x) and φ2(x) are eigenfunctions with corresponding energy eigenvalues -1 eV and -2 eV, respectively. The energy of the particle in the state ψ is

(a) – 41/25 eV
(b) – 11/25 eV
(c) – 36/5 eV
(d) – 7/5 eV

Q13. In a quantum mechanical system, an observable A is represented by an operator A. If |ψ> is a state of the system, but not an eigenstate of A, then the quantity

r = <ψ|A|ψ>2 – <ψ|A|ψ>

satisfies the relation

(a) r < 0
(b) r = 0
(c) r > 0
(d) r ≥ 0

Q3. Consider a quantum mechanical system with three linear operators A, B, and C, which are related by


where I is the unit operator. If A = d/dx and B = x, then C must be

(a) zero
(b) d/dx
(c) -x d/dx
(d) x d/dx

Q1. The wave function ψ of a quantum mechanical system described by a Hamiltonian H can be written as a Linear combination of Φ1 and Φ2 which are the eigenfunctions of H with eigenvalues E1 and E2  respectively At t = 0, the system is prepared as the state ψo = 4/5 Φ1 + 3/5 Φ2 and then allowed to evolve with time. The wavefunction at time T = 1/2 h / (E1-E2) will be (accurate to within a phase).

(a) 4/5 Φ1 + 3/5 Φ2
(b) 4/5 Φ1 + 4/5 Φ2
(c) 4/5 Φ13/5 Φ2
(d) 3/5 Φ14/5 Φ2

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Answer Key (if you find any answer wrong, feel free to Correct us)


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