1Experiment 1

The Quantum Toolkit: Operators, Eigenstates, and Measurement

Build quantum superpositions of infinite-square-well eigenstates, observe Born-rule measurement collapses, track ⟨x⟩(t), and evaluate commutators of x̂, p̂, and Ĥ.

A. Prepare a superposition

c₁1.00

c₂0.00

c₃0.00

Sliders set raw weights. The table always shows the normalised cₙ used in every calculation.

n	Norm. cₙ	\|cₙ\|² (prob.)	Eₙ
1	1.0000	1.0000	1
2	0.0000	0.0000	4
3	0.0000	0.0000	9

⟨E⟩ = 1.0000 (sim. units)

B. Wavefunction and time development

Speed1.0

t = 0.000

Re ψ(x,t) |ψ(x,t)|² ⟨x⟩(t)

⟨x⟩(t) = 0.0000(well centre = π/2 ≈ 1.5708)

Strip-chart of ⟨x⟩ while Play is running.

C. Measurement and the Born rule

Total trials: 0

E1=1

0 (0.0% — theory 100.0%)

E2=4

0 (0.0% — theory 0.0%)

E3=9

0 (0.0% — theory 0.0%)

D. Commutators

Operator A:

Operator B:

Choose two operators and click Evaluate.

Uses 4th-order finite differences for d/dx and d²/dx² on a 501-point grid. For [x̂,p̂]=iℏ (ℏ=1 here), the expectation value should be exactly i and the norm exactly 1 for any normalised state.

Observations

Record your measured data in the tables below. Refer to the lab manual for the full procedure.

Table 1.1 — Energy measurement tally (30 trials each)

Trial outcome	$c_{1} = c_{2} = 1$	$c_{1} = 2, c_{2} = 1$

Rows: E₁, E₂, E₃

Table 1.2 — Static expectation value

Quantity	Value

Row: ⟨x⟩, static c₁=c₂=1 state

Table 1.3 — Commutator results

State and commutator	Expectation value $⟨[\hat{A}, \hat{B}]⟩$	Norm $∥ [\hat{A}, \hat{B}] ψ ∥$

Rows: [Ĥ,x̂] on |1⟩; [x̂,p̂] on |1⟩; [Ĥ,x̂] on c₁=c₂=1 superposition