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Now In this particular Post of IIT JAM Physics Test Series, you will get a test of the topic General Formalism in Quantum Mechanics of Chapter Modern Physics. There are total 15 Questions given below also Answers are attached at the end of the test so that you can verify your answers after completing the test. So, Practice these Questions and Do your Best. Also Solve **IIT JAM Physics Previous Year Question Paper**. And Don’t Forget to Share with Your Friends.

**[Free] IIT JAM Physics Test Series 2023 : Modern Physics – (General Formalism in Quantum Mechanics)**

Q1. The ground state (apart from omalization) ofa particle ofunit mass moving in a one-dimensional potential V(x) is exp(-x^{2}/_{2}) cosh (√2x). The potential V(x), insuitable 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 eonjugate 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 systemis given by

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

where ‘a’ is apositive mumber 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 operator 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 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 is 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) |

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