UNIT-2 Operators in quantum mechanics, Study notes of Quantum Mechanics

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Operators in Quantum Mechanics

Operators in Quantum Mechanics 1 / 54

Outline I

(^1) Introduction

2 Core Concepts in Dirac Notation

Ket Vectors

Bra Vectors

Inner Product and Orthogonality

Outer Products and Operators

(^3) Completeness and Basis

(^4) Dirac Notation in Practice

Matrix Elements of an Operator

Expectation Values and Measurements

Example: Spin-

1

2

in Bra-Ket Notation

(^5) Numerical Example: Position Basis

6 Summary and Takeaways

(^7) Introduction

(^8) Basic Principles of Quantum Operators

(^9) Common Quantum Mechanical Operators

Outline III

Infinite Potential Well (Particle in a Box)

Quantum Harmonic Oscillator

Other Examples (Briefly)

20 Interpretation & Superposition Principle

(^21) Conclusion and Key Takeaways

Motivation and Overview

Dirac Notation (or Bra-Ket Notation) is a powerful, abstract way to

handle states and operators in quantum mechanics.

Introduced by Paul Dirac to unify the matrix and wavefunction

formulations.

Very convenient for:

Expressing quantum states in abstract Hilbert spaces.

Handling inner products, expansions in basis sets, and operators

succinctly.

Bra Vectors

A bra ⟨ϕ| is the Hermitian conjugate (dual vector) of the ket |ϕ⟩.

If |ϕ⟩ is a column vector, ⟨ϕ| is its conjugate transpose (row vector).

Symbolically,

|ϕ⟩ ↔

ϕ 1

ϕ 2

 ,^ ⟨ϕ| ↔^

ϕ

∗

1

, ϕ

∗

2

The bra acts on a ket to produce an inner product (a scalar):

⟨ϕ|ψ⟩ =

Z

ϕ

∗ (x) ψ(x) dx (in position representation).

Inner Product: ⟨ϕ|ψ⟩

The quantity ⟨ϕ|ψ⟩ is a complex scalar known as the inner product

of two states.

In wavefunction form (position representation),

⟨ϕ|ψ⟩ =

Z

ϕ

∗

(x) ψ(x) dx.

If ⟨ϕ|ψ⟩ = 0, the states are said to be orthogonal.

⟨ψ|ψ⟩ = 1 =⇒ |ψ⟩ is normalized (probability interpretation).

Physical Interpretation:

| ⟨ϕ|ψ⟩ |

2 can represent transition probabilities between states |ψ⟩ and

|ϕ⟩.

Complete Orthonormal Basis

A set of kets {|ei ⟩} forms a complete orthonormal basis if:

⟨e i

|e j

⟩ = δ ij

X

i

|e i

⟩ ⟨e i

| = ˆI ,

where ˆI is the identity operator.

Any ket |ψ⟩ can be expanded in this basis:

|ψ⟩ =

X

i

c i

|e i

⟩ , c i

= ⟨e i

|ψ⟩.

In continuous bases (e.g., position basis |x⟩), the completeness

relation becomes (^) Z

|x⟩ ⟨x| dx =

I.

Examples of Basis Sets

Position basis: |x⟩, satisfies ⟨x

′ |x⟩ = δ(x

′ − x).

Momentum basis: |p⟩, satisfies ⟨p

′ |p⟩ = δ(p

′ − p).

Energy eigenbasis: |E n

⟩, satisfies ⟨E n

|E

m

⟩ = δ nm

for discrete spectra

or δ(E

′ − E ) in the continuous case.

Spin-

1

2

states: |↑⟩ , |↓⟩, satisfies ⟨↑|↓⟩ = 0, ⟨↑|↑⟩ = 1, etc.

Expectation Values

Definition:

A⟩

ψ

= ⟨ψ|

A |ψ⟩.

Interpretation:

The average (mean) result of measuring the observable A many times

on identically prepared systems in state |ψ⟩.

If |ai ⟩ are eigenstates of ˆA with eigenvalues ai , then

A⟩ψ =

X

i

|ci |

2

ai , ci = ⟨ai |ψ|ai |ψ⟩.

Spin-

1

2

Example

Basis Kets: |↑⟩ and |↓⟩ along z-axis:

S

z

S

z

A general spin-

1

2

state:

|ψ⟩ = α |↑⟩ + β |↓⟩ , (|α|

2

• |β|

2 = 1).

Measurement Probability of

S

z

P(↑) = |α|

2 , P(↓) = |β|

2 .

Expectation Value:

S

z

⟩ = ⟨ψ|

S

z

|ψ⟩ =

(|α|

2

− |β|

2

).

Key Takeaways

1 Bra-Ket Notation offers a clean, abstract way to handle quantum

states without immediately specifying a representation.

(^2) Inner Products (bras acting on kets) give transition amplitudes and

probabilities.

(^3) Operators can be built via outer products or examined in a chosen

basis through matrix elements.

(^4) Completeness Relations and expansions in orthonormal bases are

central to solving QM problems.

5 Applicability: Any quantum system—spin, position, energy

eigenstates—can be expressed in Dirac notation.

Further Remarks

Dirac notation is foundational to understand advanced topics such as:

Perturbation theory

Quantum field theory

Quantum computing (qubits, gates, etc.)

Mastering bra-ket notation simplifies manipulations and clarifies the

physics behind the math.

Postulates and Properties

Postulate: Every physical observable A is represented by a linear,

Hermitian operator ˆA on the Hilbert space.

Linearity:

A(αψ + βϕ) = α

Aψ + β

Aϕ.

Hermiticity (Self-Adjointness):

⟨ϕ|

Aψ⟩ = ⟨

Aϕ|ψ⟩.

Commutation Relations:

[ ˆA,

B] = ˆA

B −

B

A.

If [ ˆA,

B] = 0, the observables A and B are said to be compatible.

Physical Meaning of Operators

The eigenvalue equation for an operator ˆA is:

A |a⟩ = a |a⟩ ,

where |a⟩ is the eigenstate with eigenvalue a.

Measurement of the observable A in the state |a⟩ yields the value a

with probability 1.

General state |ψ⟩ can be expanded in the basis of eigenstates of ˆA.

The probability of measuring a particular eigenvalue ai is given by

|⟨a i

|ψ⟩|

2 .