# Quantum Mechanics [GATE (Graduate Aptitude Test in Engineering) Physics]: Questions 26 - 31 of 80

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## Question 26

Appeared in Year: *2011*

### Question

MCQ▾The normalized eigen states of a particle in a one – dimensional potential well

are given by, , where

The particle is subjected to a perturbation,

The shift in the ground state energy due to the perturbation, in the first order perturbation theory, is –

### Choices

Choice (4) | Response | |
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a. | ||

b. | ||

c. | ||

d. |

### Passage

To the given unperturbed Hamiltonian , we add a small perturbation given by, ; where is a small quantity.

## Question 27 (1 of 2 Based on Passage)

Appeared in Year: *2013*

### Question

MCQ▾A pair of eigenvalues of the perturbed Hamiltonian, using first order perturbation theory, is –

### Choices

Choice (4) | Response | |
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a. | ||

b. | ||

c. | ||

d. |

## Question 28 (2 of 2 Based on Passage)

Appeared in Year: *2013*

### Question

MCQ▾The ground state eigenvector of the unperturbed Hamiltonian is –

### Choices

Choice (4) | Response | |
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a. | ||

b. | ||

c. | ||

d. |

## Question 29

Appeared in Year: *2013*

### Question

MCQ▾Which one of the following commutation relation is NOT CORRECT? Here, symbols have their usual meanings.

### Choices

Choice (4) | Response | |
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a. | ||

b. | ||

c. | ||

d. |

## Question 30

Appeared in Year: *2013*

### Write in Short

Short Answer▾Consider the wave function , where is the normalization constant. For , the magnitude of probability current density up to two decimal places, in unit of , is ________.

## Question 31

Appeared in Year: *2013*

### Write in Short

Short Answer▾A spin – half particle is in a linear superposition of its spin – up and spin – down states. If and are the eigenstate of then what is the expectation value, up to one decimal place, of the operator ? Here, symbols have their usual meaning.