Quantum Physics Foundations: Simple Quantum States in One Dimension

This chapter builds a minimal toolbox for working with simple one-dimensional quantum states using the wavefunction as a practical representation. The goal is to make predictions about position measurements (where the particle is likely to be found) without needing to solve differential equations in detail.

1) The wavefunction as a state representation (and what is observable)

In one dimension, a quantum state can be represented by a complex-valued function ψ(x) called the wavefunction. Think of ψ(x) as a compact encoding of everything the state can predict about position measurements.

Probability density: |ψ(x)|²

The directly testable prediction for position is the probability density:

ρ(x) = |ψ(x)|² = ψ*(x)ψ(x)

Interpretation: if you repeat the same experiment many times on identically prepared systems, then the probability of finding the particle between x and x + dx is approximately:

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P(x to x+dx) ≈ |ψ(x)|² dx

• Observable (for position): the distribution of outcomes across many trials, summarized by |ψ(x)|².
• Not directly observable: the wavefunction itself (including its overall complex phase). You infer ψ indirectly from measurement statistics and how the state responds to operations.

What about the sign or complex phase?

Two different wavefunctions can have the same |ψ(x)|² but differ in phase. For predicting position alone at a single instant, |ψ|² is the key. Phase becomes crucial when you combine alternatives or compare regions (for example, when states overlap), but you can still do a lot of position reasoning just from the shape of |ψ|².

2) Normalization and boundary conditions: constraints on allowed states

Normalization: total probability must be 1

If the particle must be found somewhere on the line, the total probability over all positions must be 1:

∫_{-∞}^{∞} |ψ(x)|² dx = 1

This is not optional: it is the mathematical statement that “one of the possible positions will occur.”

Step-by-step: normalizing a simple piecewise wavefunction

Suppose a state is uniform inside a region and zero outside (a crude “boxy” packet):

ψ(x) = A for -L/2 ≤ x ≤ L/2, and ψ(x)=0 otherwise.

To normalize:

1. Compute |ψ(x)|²: inside the region it is |A|², outside it is 0.
2. Integrate: ∫ |ψ|² dx = ∫_{-L/2}^{L/2} |A|² dx = |A|² L.
3. Set equal to 1: |A|² L = 1.
4. Choose A: a common choice is real and positive, A = 1/√L (any overall phase would also satisfy normalization).

Boundary conditions: the physical setup restricts the shape

Boundary conditions encode the constraints imposed by the environment or idealizations of it. They are not “extra math rules”; they represent the physical statement “the state must behave like this at the edges.” Common examples in 1D:

• Hard wall (impenetrable boundary): the probability of finding the particle beyond the wall is zero, so the wavefunction is taken to vanish at the wall: ψ(x_wall)=0.
• Confined region: outside a certain region, ψ(x) may be zero (idealized) or rapidly decaying (more realistic).
• Free space: no special boundary at finite x; instead you often use idealized states that extend indefinitely.

In this chapter, you will use boundary conditions mainly as a way to recognize which “archetype” of state is appropriate for a situation.

3) Three archetypal 1D states (shapes you should recognize)

Many realistic states can be approximated as combinations of a few common shapes. Here are three archetypes and when they are useful.

A) Localized wave packet (particle likely in a region)

A localized wave packet has |ψ(x)|² concentrated in a finite region and small elsewhere. A common mental picture is a bell-shaped curve (Gaussian-like), though many localized shapes exist.

What it predicts: position measurements are most likely near the peak of |ψ|², with decreasing likelihood away from it.

Physical situations where it applies:

• A particle prepared by a narrow slit or aperture (position selection).
• A particle trapped temporarily in a region and then released.
• An electron in a localized region of a device where you can say “it’s around here” with some spread.

Practical reading rule: the narrower the packet in x, the more sharply peaked the position distribution is (more “localized”).

B) Standing-wave-like bound state (confined with nodes)

A standing-wave-like bound state is typical when the particle is confined between boundaries. The wavefunction has a structured pattern with nodes (points where ψ(x)=0) and antinodes (where |ψ| is large).

What it predicts: there are positions that are never observed (nodes), and positions that are more likely (near antinodes). The probability density often has multiple “lobes.”

Physical situations where it applies:

• A particle confined between hard walls (idealized 1D “box”).
• Any strongly confining setup where the particle remains in a region for long times and the state is stationary-like.

Boundary-condition signature: if there are hard walls at x=0 and x=L, then ψ(0)=ψ(L)=0, which naturally leads to standing-wave-like shapes.

C) Delocalized plane-wave-like state (uniform probability density)

A plane-wave-like state extends across space with nearly constant magnitude. In its ideal form, |ψ(x)|² is constant, meaning every position is equally likely.

Important caveat: an exactly constant |ψ|² over all space cannot be normalized in the usual way (the integral would be infinite). In practice, you use it as an idealization for a particle that is spread out over a very large region, or you use a “box normalization” (treating space as a large finite interval) to make calculations manageable.

What it predicts (in the idealized sense): no preferred position; the particle is not localized.

Physical situations where it applies:

• A particle in a long, uniform region where no location is special (approximate translational symmetry).
• As a building block for more realistic states: localized packets can be constructed as superpositions of plane-wave-like components.

Practical reading rule: if |ψ|² is flat (or nearly flat) across the region of interest, position measurements will be broadly spread with no favored location.

4) Expectation value: the average of many trials (not a single outcome)

When you measure position many times on identically prepared systems, you get a distribution of outcomes. The expectation value of position, written ⟨x⟩, is the average outcome you would get in the long run.

Conceptual definition

Imagine repeating the experiment N times and recording outcomes x₁, x₂, …, x_N. The sample average is:

x̄ = (1/N) Σ x_i

As N becomes very large, x̄ approaches the expectation value ⟨x⟩ predicted by the state.

Wavefunction formula (for position)

Using the probability density |ψ(x)|²:

⟨x⟩ = ∫ x |ψ(x)|² dx

This is a weighted average: positions with larger |ψ|² contribute more to the average.

How expectation differs from a single outcome

• Single outcome: one measurement returns one value (one point on the line). It can land away from the peak, especially if the distribution has long tails.
• Expectation value: a property of the distribution; it summarizes the “center of mass” of |ψ|².

Example (qualitative): if |ψ|² is symmetric about x=0, then ⟨x⟩=0 even though individual measurements can be positive or negative.

5) Practice activities: reading ψ sketches and predicting outcomes

In these activities, you will treat sketches of ψ(x) (or directly of |ψ(x)|²) as prediction tools. The key habit: always translate “shape of ψ” into “shape of |ψ|²,” then read probabilities from the area under the curve.

Activity 1: Identify likely regions from a localized packet

Sketch description: |ψ(x)|² is a single hump centered at x=a, small elsewhere.

• Question: Where is the particle most likely to be found?
• Step-by-step:
  1. Locate the maximum of |ψ|² (the peak).
  2. State that outcomes cluster near that x value.
  3. Identify an interval around the peak where most of the area lies (e.g., “between a-Δ and a+Δ”).
• Prediction: Most measurements land near x=a; measurements far away are rare because the density there is small.

Activity 2: Compare two packets with different widths

Sketch description: Packet A is narrow and tall; Packet B is wider and shorter, both normalized.

• Question: Which state gives more precise position outcomes?
• Step-by-step:
  1. Remember normalization: both have total area 1 under |ψ|².
  2. A narrower distribution means outcomes are concentrated in a smaller range of x.
  3. A wider distribution means outcomes are spread out across a larger range.
• Prediction: Packet A yields more tightly clustered position results; Packet B yields more spread-out results.

Activity 3: Read nodes and lobes in a standing-wave-like bound state

Sketch description: ψ(x) oscillates between x=0 and x=L and touches zero at the boundaries; it also crosses zero at one interior point (one node inside).

• Question: Are there positions where the particle is never found?
• Step-by-step:
  1. Convert to probability density: |ψ|² is zero wherever ψ=0.
  2. Mark the node locations (including boundaries if applicable).
  3. Identify lobes between nodes where |ψ|² is nonzero.
• Prediction: Measurements never return the node positions; outcomes cluster in the lobes (near antinodes).

Activity 4: Symmetry and expectation value

Sketch description: |ψ(x)|² has two equal peaks at x=+a and x=-a, symmetric about 0.

• Question: What is ⟨x⟩ likely to be, and where do individual outcomes land?
• Step-by-step:
  1. Use symmetry: equal weight on both sides implies the weighted average cancels.
  2. Note that the most likely outcomes are near the peaks, not necessarily near the average.
• Prediction: ⟨x⟩ = 0, but single measurements tend to land near +a or -a.

Activity 5: Plane-wave-like state in a finite region (box-normalized intuition)

Sketch description: |ψ(x)|² is approximately constant from x=0 to x=L and zero outside (an idealized “uniform in a region” state).

• Question: What is the probability of finding the particle in the left half versus the right half?
• Step-by-step:
  1. Constant density means probability is proportional to interval length.
  2. Left half length: L/2; right half length: L/2.
  3. Equal lengths imply equal probabilities.
• Prediction: P(0 to L/2) = P(L/2 to L) = 1/2.

Activity 6: How changing the sketch changes predictions (area thinking)

Rule of thumb: probabilities come from areas under |ψ|² over intervals.

• Change 1: shift the whole distribution right by +d
  • Effect: all “likely positions” shift right; ⟨x⟩ increases by d.
• Change 2: make one lobe taller while keeping normalization
  • Effect: more area moves into that region, increasing the probability of outcomes there; other regions must lose area to keep total probability 1.
• Change 3: introduce a node at a point
  • Effect: probability at that exact point becomes zero; the distribution splits into separated regions of support.

Quick reference table: what to read off a ψ sketch