Magnetic translation

Key takeaways

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

Reference excerpt

In quantum mechanics, the action of symmetries on physical states are represented by either linear or, more generally, projective representations. For a particle moving in a crystal without a magnetic field, spatial translations are represented linearly and the corresponding translation operators commute with one another and with the Hamiltonian. However, in the presence of a magnetic field, even when the magnetic field configuration is translationally invariant, wave functions fail to transform linearly under translation. Instead, they are represented projectively, acquiring position-dependent phase factors. The resulting operators are known as magnetic translation operator .

Magnetic Symmetry Operator In this section, we will start with the discussion of the more well-known magnetic translation operator and generalize it to other spatial symmetries such as rotations.

Magnetic Translation Operator To be more specific, consider the Hamiltonian of a quantum particle (with charge q {\displaystyle q} and mass m {\displaystyle m} ) in a magnetic field explicitly depends on the magnetic vector potential A ( r ) {\displaystyle {\bf {A}}({\bf {r}})} , where B = ∇ × A ( r ) {\displaystyle {\bf {B}}=\nabla \times {\bf {A}}({\bf {r}})} :

H = [ p − q A ( r ) ] 2 2 m + V ( r ) {\displaystyle H={\frac {[{\bf {p}}-q{\bf {A}}({\bf {r}})]^{2}}{2m}}+V({\bf {r}})} . For a uniform magnetic field B ( r ) = B z ^ {\displaystyle {\bf {B}}({\bf {r}})=B{\hat {z}}} , the ordinary translation operator T ( a ) = e − i p ⋅ a / ℏ {\displaystyle T({\bf {a}})=e^{-i{\bf {p}}\cdot {\bf {a}}/\hbar }} does not commute with the Hamiltonian even when V ( r + a ) = V ( r ) {\displaystyle V({\bf {r+a}})=V({\bf {r}})} because

T ( a ) − 1 [ p − q A ( r ) ] T ( a ) = p − q A ( r + a ) = p − q A ( r ) − q [ A ( r + a ) − A ( r ) ] = p − q A ( r ) − q ∇ χ a ( r ) . {\displaystyle T({\bf {a}})^{-1}[{\bf {p}}-q{\bf {A}}({\bf {r}})]T({\bf {a}})={\bf {p}}-q{\bf {A}}({\bf {r+a}})={\bf {p}}-q{\bf {A}}({\bf {r}})-q[{\bf {A}}({\bf {r+a}})-{\bf {A}}({\bf {r}})]={\bf {p}}-q{\bf {A}}({\bf {r}})-q\nabla \chi _{\bf {a}}({\bf {r}}).} In the third equality, one writes A ( r + a ) − A ( r ) = ∇ χ a ( r ) {\displaystyle {\bf {A}}({\bf {r+a}})-{\bf {A}}({\bf {r}})=\nabla \chi _{\bf {a}}({\bf {r}})} using the fact that

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Worked examples

Example 1 — a first encounter with Magnetic translation

Start with the simplest possible case. Write down what Magnetic translation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, 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 Magnetic translation 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 Magnetic translation 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 Magnetic translation

In research: Magnetic translation appears in physics 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 Magnetic translation 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: Magnetic translation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Magnetism, Mathematical physics stubs, Quantum magnetism, so understanding it makes those chapters shorter.
In everyday life: Look for Magnetic translation 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.

How to study Magnetic translation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Magnetic translation 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 Magnetic translation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Magnetic translation in simple terms?

In quantum mechanics, the action of symmetries on physical states are represented by either linear or, more generally, projective representations. For a particle moving in a crystal without a magnetic field, spatial translations are represented linearly and the corresponding translation operators c…

Why does Magnetic translation matter?

Because it connects several physics 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 Magnetic translation?

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 Magnetic translation.