**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) x^{2}/_{2}

(b) x^{2}/_{2} – √2 x tanh(√2x)

(c) x^{2}/_{2} – √2 x tan(√2x)

(d) x^{2}/_{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 e^{ikx/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) ^{ℏ}/_{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 = (ℏ^{2}λ^{2})/_{m}

(b) E = -(ℏ^{2}λ^{2})/_{m}

(c) E = -(ℏ^{2}λ^{2})/_{m}

(d) E = (ℏ^{2}λ^{2})/_{m}

Q8. Given that ψ1 and ψ2 are eigenstates of a Hamiltonian with eigenvalues E_{1} and E_{2} respectively, what is the energy uncertainty in the state (ψ1 + ψ_{2})?

(a) -√(E_{1}E_{2})

(b) ^{1}/_{2} |E_{1} – E_{2}|

(c) ^{1}/_{2} |E_{1} + E_{2}|

(d) ^{1}/_{√2} |E_{2} – E_{1}|

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(a^{ea }+ x)

(c) ψ(e^{a} x)

(d) e^{a} ψ(x)

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

ψ(x) = ^{3}/_{5} φ_{1}(x) + ^{4}/_{5} φ_{2}(x)

where φ_{1}(x) and φ_{2}(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

**, then the quantity**

*A*r = <ψ|* A*|ψ>

^{2}– <ψ|

**|ψ>**

*A*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

**, which are related by**

*C*** AB** –

**=**

*C*

*I*where ** I** is the unit operator. If

**=**

*A*^{d}/

_{dx}and

**= x, then**

*B**must be*

**C**(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 Φ

_{1}and Φ

_{2}which are the eigenfunctions of

**with eigenvalues E**

*H*_{1}and E

_{2}respectively At t = 0, the system is prepared as the state ψ

_{o}=

^{4}/

_{5}Φ

_{1}+

^{3}/

_{5}Φ

_{2}and then allowed to evolve with time. The wavefunction at time T =

^{1}/

_{2}h / (E

_{1}-E

_{2}) will be (accurate to within a phase).

(a) ^{4}/5 Φ_{1} + ^{3}/_{5} Φ_{2}

(b) ^{4}/_{5} Φ_{1} + ^{4}/_{5} Φ_{2}

(c) ^{4}/_{5} Φ_{1} – ^{3}/_{5} Φ_{2}

(d) ^{3}/_{5} Φ_{1} – ^{4}/_{5} Φ_{2}

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

*Answer Key*

01. | (b) | 06. | (b) | 11. | (c) |

02. | (d) | 07. | (c) | 12. | (a) |

03. | (a) | 08. | (b) | 13. | (a) |

04. | (c) | 09. | (c) | 14. | (d) |

05. | (b) | 10. | (c) | 15. | (c) |