Quantum Mechanics II & Atomic Physics-Quantum Mechanics II [Optionals IAS Mains Physics]: Questions 1 - 6 of 23

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Question 1

Quantum Mechanics II
The Spin Half Problem and Properties of Pauli Spin Matrices

Appeared in Year: 2011 (IFS)

Describe in Detail

Essay▾

What are Pauli spin matrices?

Show that:

Where, are Pauli spin matrices and and are vector operators which commute with , but do not necessarily commute with each other?

Explanation

Pauli spin matrices or Pauli matrices are a set of matrices, which are complex, Hermitian and unitary matrices. Pauli spin matrices defined as,

The Pauli vector is defined by,

From question another vector operators are and are given.

Now, mapping mechanism from a Pauli matrix basis and a vector operator basis are provided as,

Similarly,

From commutat…

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Question 2

Quantum Mechanics II
The Spin Half Problem and Properties of Pauli Spin Matrices
Edit

Appeared in Year: 2003

Write in Short

Short Answer▾

Show that the Pauli Matrices anti commute. (5 marks)

Question 3

Quantum Mechanics II
The Angular Momentum Problem

Appeared in Year: 2003

Describe in Detail

Essay▾

Express the Cartesian components of the angular momentum L in operator form. Show that . What is the significance of this commutation relation? (15 Marks)

Explanation

The orbital angular momentum can be obtained at once by replacing by the corresponding operators in the position representation,

The Cartesian components of are

Clearly, angular momentum doesn՚t exist in a one dimensional space.

The importance of this observable is that it is compatible with any of the Cartesian components of the angular momentum

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Question 4

Quantum Mechanics II
The Hydrogen Atom

Appeared in Year: 2003

Write in Short

Short Answer▾

Show that the radial probability density of the ground state of the hydrogen atom how a maximum at . The ground state wave function of the hydrogen atom is given by Where a is the Bohr Radius (10 Marks)

Question 5

Quantum Mechanics II
The Hydrogen Atom

Write in Short

Short Answer▾

The energy levels of a hydrogen atom are given by where Show that

Question 6

Quantum Mechanics II
The Hydrogen Atom

Appeared in Year: 2009

Describe in Detail

Essay▾

The quantum mechanical probability distribution function of an electron of ground state of hydrogen atom is

Using the result deduct that N is proportional to . (20 Marks)

Explanation

Applying integration by parts on the above integral

It is clear that exponential will “beat” any power function as of tends to .

Hence N is proportional to .

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