Quantum Physics Foundations: Superposition as a Physical State Description

1) Superposition as a statement about the quantum state

In quantum physics, a superposition is a precise way to describe a system’s state when it is not in a single basis state for the measurement you have in mind. Mathematically, it means the state can be written as a linear combination of basis states.

For a two-outcome measurement with basis states |0⟩ and |1⟩, a general state is written

|ψ⟩ = α|0⟩ + β|1⟩

Here α and β are complex numbers called probability amplitudes. The basis states |0⟩ and |1⟩ represent the two possible outcomes of that measurement. The key physical statement is: if you measure in that basis, the outcomes are generally probabilistic, with probabilities determined by the amplitudes.

Accessible analogy (and where it stops)

Analogy: Think of two musical modes on a string: a “mode A” and “mode B.” You can excite a combination of both modes at once. The overall vibration is a sum of contributions from each mode.

Physics (not the analogy): In quantum mechanics, the “sum” is not about two classical things coexisting in space. It is a statement about the system’s state vector and how it will produce outcomes when you choose a measurement. The basis states are tied to a measurement setting, not to “what the system really is doing” in a classical sense between measurements.

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2) Two-state systems and state vectors (without picturing a physical arrow)

Many foundational examples reduce to a two-state system: polarization of a photon (two orthogonal polarizations), a spin-1/2-like degree of freedom (two outcomes along a chosen axis), or any setup with two exclusive outcomes.

Conceptually, we represent the state as a vector in an abstract two-dimensional space (a Hilbert space). Using a chosen basis {|0⟩, |1⟩}, we can represent

|ψ⟩ ↔ ( α , β )

This looks like an ordinary vector, but it is crucial not to misinterpret it:

• It is not a physical arrow in 3D space. The components (α, β) are coordinates relative to a chosen measurement basis.
• Changing the basis changes the coordinates. The same physical state can look like different pairs (α, β) depending on which measurement you are describing.
• Overall phase is not directly observable. Multiplying the whole state by e^{iφ} does not change measurement probabilities in a fixed basis; what matters are relative phases between components (how the components relate to each other).

Definite vs probabilistic outcomes in a given basis

If the state equals one of the basis states, the outcome is definite in that basis:

• If |ψ⟩ = |0⟩, then measuring in the {|0⟩, |1⟩} basis yields outcome 0 with probability 1.
• If |ψ⟩ = |1⟩, then outcome 1 occurs with probability 1.

If both coefficients are nonzero, the outcomes are generally probabilistic. (The exact probabilities depend on the amplitudes; you do not need to picture the system as “partly 0 and partly 1” in a classical mixture sense.)

3) Changing the measurement setting changes the basis states

“Superposition” is not an absolute label; it is always relative to a chosen measurement. A state can be a basis state (definite) for one measurement setting and a superposition (probabilistic) for another.

Suppose you have two different measurement settings, each with two outcomes:

• Setting A has basis {|0⟩, |1⟩}
• Setting B has basis {|+⟩, |−⟩}

These bases are related by a change of coordinates. A common example is:

|+⟩ = (|0⟩ + |1⟩)/√2   and   |−⟩ = (|0⟩ − |1⟩)/√2

Now notice what this means physically:

• If the system is prepared in |0⟩, then it is definite for measurement A.
• But in measurement B, the same state |0⟩ can be rewritten as |0⟩ = (|+⟩ + |−⟩)/√2, which is a superposition of B’s outcomes.

So the statement “the system is in a superposition” really means: relative to the measurement you are about to perform, the state is not one of that measurement’s basis states.

Concrete example: polarization bases

Let |H⟩ and |V⟩ be horizontal and vertical polarization outcomes for a polarizer aligned horizontally/vertically. Let |D⟩ and |A⟩ be diagonal/anti-diagonal outcomes for a polarizer rotated by 45°.

A standard relationship is:

|D⟩ = (|H⟩ + |V⟩)/√2   and   |A⟩ = (|H⟩ − |V⟩)/√2

If a photon is prepared as |H⟩, then:

• Measuring with an H/V polarizer: outcome H is definite.
• Measuring with a D/A polarizer: outcomes D and A are probabilistic (in fact, equal in this symmetric case).

4) Practice: decide what is definite vs probabilistic from a preparation

The skill you want is: given a preparation description, identify which measurement settings have definite outcomes and which have probabilistic outcomes.

Practice method (step-by-step)

• Step 1: Identify the preparation basis. What measurement setting would make the preparation an eigenstate (a basis state)?
• Step 2: Identify the measurement basis you will use. What are the two possible outcomes for that measurement?
• Step 3: Compare bases. If the measurement basis matches the preparation basis, the outcome is definite. If it is rotated/changed, rewrite the prepared state in the measurement basis (conceptually) to see it as a superposition.
• Step 4: Decide “definite vs probabilistic.” If the rewritten state has only one basis component, definite; if it has two nonzero components, probabilistic.

Exercise 1: photon through polarizers

Preparation: A photon passes through a polarizer oriented at 0° (horizontal). After the polarizer, the photon is in state |H⟩.

Question A: If you measure with another polarizer at 0° (H/V basis), which outcomes are definite/probabilistic?

• Because the measurement basis matches the preparation basis, the outcome is definite: H occurs with certainty; V is excluded.

Question B: If you measure with a polarizer at 45° (D/A basis), which outcomes are definite/probabilistic?

• The measurement basis differs from the preparation basis, so |H⟩ is a superposition of |D⟩ and |A⟩. Outcomes are probabilistic: you can get D or A.

Question C: If you measure with a polarizer at 90° (V), which outcomes are definite/probabilistic?

• In the H/V basis, 90° corresponds to V. Since the state is |H⟩, the outcome is still definite in the sense that H is certain and V is impossible (assuming ideal polarizers and no loss channels).

Exercise 2: “spin-like” preparation and measurement axis

Preparation: A two-state system is prepared to be “up” along the z-setting, i.e., |z+⟩.

Question: You now measure along the x-setting with outcomes |x+⟩ and |x−⟩. Which outcomes are definite/probabilistic?

• The measurement setting changed (z to x), so |z+⟩ is a superposition of |x+⟩ and |x−⟩. Therefore the outcomes are probabilistic in the x-measurement.

Exercise 3: identify the basis where the outcome is definite

Preparation: A photon is prepared by passing through a 45° polarizer (diagonal). This prepares |D⟩.

Task: For each measurement setting, label outcomes as definite or probabilistic:

• D/A measurement (45°): definite (D certain, A excluded).
• H/V measurement (0°/90°): probabilistic (H and V possible).

5) What superposition is not

Superposition is often misunderstood as if it were just “we don’t know which definite state the system is really in.” That is classical ignorance (a mixture), and it is not what a pure quantum superposition means.

Careful comparison: mixture vs superposition (without overreaching)

A practical way to keep the distinction straight is to focus on what is being asserted:

• Superposition is a state description. It specifies how to compute outcome probabilities for different measurement settings.
• It is not a claim that two classical alternatives are simultaneously true. The basis states are tied to a measurement context.
• It is not merely ignorance about a hidden definite value in the same basis. If you treat a superposition as “really one outcome but unknown,” you generally fail to account for how changing the measurement basis changes what is definite vs probabilistic for the same preparation.

Common “not this” statements (with corrected versions)

• Not: “The system is partly in |0⟩ and partly in |1⟩ like paint mixed from two colors.” Instead: The state is a linear combination of basis states, used to predict measurement outcomes in that basis.
• Not: “The state vector points somewhere in physical space.” Instead: It is an abstract vector; its components depend on the chosen basis (measurement setting).
• Not: “Superposition means we just don’t know the real outcome.” Instead: Superposition is a specific quantum state that can be definite in one basis and probabilistic in another, reflecting the structure of quantum measurements rather than ordinary ignorance.