science

Schwinger variational principle

Schwinger variational principle is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Schwinger variational principle rather than just read about it. In short: Schwinger variational principle is a variational principle which expresses the scattering T-matrix as a functional depending on two unknown wave functions. The functional attains stationary value equal to actual scattering T-matrix.

Key takeaways

Schwinger variational principle belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Schwinger variational principle to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Schwinger variational principle from memory before moving on to harder problems.

Reference excerpt

Schwinger variational principle is a variational principle which expresses the scattering T-matrix as a functional depending on two unknown wave functions. The functional attains stationary value equal to actual scattering T-matrix. The functional is stationary if and only if the two functions satisfy the Lippmann-Schwinger equation. The development of the variational formulation of the scattering theory can be traced to works of L. Hultén and J. Schwinger in 1940s.

Linear form of the functional The T-matrix expressed in the form of stationary value of the functional reads

⟨ ϕ ′ | T ( E ) | ϕ ⟩ = T [ ψ ′ , ψ ] ≡ ⟨ ψ ′ | V | ϕ ⟩ + ⟨ ϕ ′ | V | ψ ⟩ − ⟨ ψ ′ | V − V G 0 ( + ) ( E ) V | ψ ⟩ , {\displaystyle \langle \phi '|T(E)|\phi \rangle =T[\psi ',\psi ]\equiv \langle \psi '|V|\phi \rangle +\langle \phi '|V|\psi \rangle -\langle \psi '|V-VG_{0}^{(+)}(E)V|\psi \rangle ,}

where ϕ {\displaystyle \phi } and ϕ ′ {\displaystyle \phi '} are the initial and the final states respectively, V {\displaystyle V} is the interaction potential and G 0 ( + ) ( E ) {\displaystyle G_{0}^{(+)}(E)} is the retarded Green's operator for collision energy E {\displaystyle E} . The condition for the stationary value of the functional is that the functions ψ {\displaystyle \psi } and ψ ′ {\displaystyle \psi '} satisfy the Lippmann-Schwinger equation

| ψ ⟩ = | ϕ ⟩ + G 0 ( + ) ( E ) V | ψ ⟩ {\displaystyle |\psi \rangle =|\phi \rangle +G_{0}^{(+)}(E)V|\psi \rangle }

and

| ψ ′ ⟩ = | ϕ ′ ⟩ + G 0 ( − ) ( E ) V | ψ ′ ⟩ . {\displaystyle |\psi '\rangle =|\phi '\rangle +G_{0}^{(-)}(E)V|\psi '\rangle .}

Fractional form of the functional Different form of the stationary principle for T-matrix reads

⟨ ϕ ′ | T ( E ) | ϕ ⟩ = T [ ψ ′ , ψ ] ≡ ⟨ ψ ′ | V | ϕ ⟩ ⟨ ϕ ′ | V | ψ ⟩ ⟨ ψ ′ | ( V − V G 0 ( + ) ( E ) V ) | ψ ⟩ . {\displaystyle \langle \phi '|T(E)|\phi \rangle =T[\psi ',\psi ]\equiv {\frac {\langle \psi '|V|\phi \rangle \langle \phi '|V|\psi \rangle }{\langle \psi '|(V-VG_{0}^{(+)}(E)V)|\psi \rangle }}.}

The wave functions ψ {\displaystyle \psi } and ψ ′ {\displaystyle \psi '} must satisfy the same Lippmann-Schwinger equations to get the stationary value.

Application of the principle The principle may be used for the calculation of the scattering amplitude in the similar way like the variational principle for bound states, i.e. the form of the wave functions ψ , ψ ′ {\displaystyle \psi ,\psi '} is guessed, with some free parameters, that are determined from the condition of stationarity of the functional.

See also Lippmann–Schwinger equation Quantum scattering theory T-matrix method Green's operator

References

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Worked examples

Example 1 — a first encounter with Schwinger variational principle

Start with the simplest possible case. Write down what Schwinger variational principle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Schwinger variational principle before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Schwinger variational principle ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Schwinger variational principle

In research: Schwinger variational principle appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Schwinger variational principle in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Schwinger variational principle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Scattering, Scattering stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Schwinger variational principle outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Schwinger variational principle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Schwinger variational principle means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Schwinger variational principle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Schwinger variational principle in simple terms?

Why does Schwinger variational principle matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Schwinger variational principle?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Schwinger variational principle.