Back to Exercises Topic: canonical quantization

Sheet 1

Canonical Quantization Foundations

Operator replacement, commutators, and ladder-operator intuition for free fields.

hamiltonian field theory canonical quantization particles and fields

Exercise 1

From classical variables to operators

Intermediatecanonical quantizationcommutatorsscalar fields

Starting from a real scalar field with canonical pair $\phi(x)$ and $\pi(x)$ , explain the formal replacement that leads to canonical quantization.

Subparts

Write the equal-time commutator imposed after quantization.

State in one sentence how this differs conceptually from a classical Poisson bracket.

Exercise 2

Mode-by-mode oscillator picture

Intermediatefourier modesharmonic oscillatorparticle interpretation

A free scalar field decomposes into independent Fourier modes. Explain why each mode behaves like a harmonic oscillator.

Subparts

What parameter plays the role of the oscillator frequency for momentum mode $\mathbf{k}$?

Why does this make the particle interpretation natural?

Exercise 3

Vacuum and one-particle states

Beginnervacuum stateladder operatorsfock space

Define the vacuum and one-particle states in the oscillator language of a free field.

Subparts

What equation defines the vacuum?

How is a one-particle state with momentum $\mathbf{k}$ constructed?

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Back to Exercises Topic: canonical quantization

Sheet 1

Canonical Quantization Foundations

Operator replacement, commutators, and ladder-operator intuition for free fields.

hamiltonian field theory canonical quantization particles and fields

Exercise 1

From classical variables to operators

Intermediatecanonical quantizationcommutatorsscalar fields

Starting from a real scalar field with canonical pair $\phi(x)$ and $\pi(x)$ , explain the formal replacement that leads to canonical quantization.

Subparts

Write the equal-time commutator imposed after quantization.

State in one sentence how this differs conceptually from a classical Poisson bracket.

Exercise 2

Mode-by-mode oscillator picture

Intermediatefourier modesharmonic oscillatorparticle interpretation

A free scalar field decomposes into independent Fourier modes. Explain why each mode behaves like a harmonic oscillator.

Subparts

What parameter plays the role of the oscillator frequency for momentum mode $\mathbf{k}$?

Why does this make the particle interpretation natural?

Exercise 3

Vacuum and one-particle states

Beginnervacuum stateladder operatorsfock space

Define the vacuum and one-particle states in the oscillator language of a free field.

Subparts

What equation defines the vacuum?

How is a one-particle state with momentum $\mathbf{k}$ constructed?

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Next sheet

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