Wavefunction Expectation Value Calculator

Wavefunction Expectation Value Calculator | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Science & Education
Input types: textarea
Output type: text
Sample coverage: 4
API ready: Yes

Overview

The Wavefunction Expectation Value Calculator computes quantum statistical properties from sampled real wavefunctions ψ(x). Using trapezoidal numerical integration, it calculates the normalization integral ⟨ψ|ψ⟩ = ∫ψ²dx, position expectation values ⟨x⟩ and ⟨x²⟩, and position uncertainty σ_x = √(⟨x²⟩ − ⟨x⟩²) from coordinate-amplitude pairs.

When to use

Analyzing numerical or discretized wavefunction outputs from quantum mechanical simulations and Schrödinger solvers.
Evaluating normalization constants and spatial probability spreads for sampled real-valued wavepackets.
Verifying quantum homework problems, state symmetries, and position expectation values across 1D potentials.

How it works

1Paste discrete real wavefunction data as one coordinate and amplitude pair (x ψ(x)) per line.
2The tool parses the data and automatically sorts all distinct coordinate points in ascending order.
3Using the trapezoidal rule, it integrates the inner product ⟨ψ|ψ⟩ = ∫ψ²dx as well as weighted integrals ∫x·ψ²dx and ∫x²·ψ²dx.
4It outputs the total normalization value, normalized position expectation values ⟨x⟩ and ⟨x²⟩, and standard deviation σ_x.

Use cases

Calculating the center of mass ⟨x⟩ and spatial spread σ_x for numerical harmonic oscillator ground and excited states.
Checking whether truncated numerical grids adequately capture wavepacket probability density through the normalization integral ⟨ψ|ψ⟩.
Demonstrating expectation value shifts and wavepacket displacement across shifted Gaussian distributions.

Examples

1. Symmetric Gaussian Wavepacket

Physics Student

Background

Analyzing a centered, unnormalized Gaussian ground state distribution ψ(x) = exp(−x²/2) over the interval [-3, 3].

Problem

Determine the normalization inner product and confirm that position expectation ⟨x⟩ is zero due to symmetry.

How to use

Paste 13 sampled (x, ψ(x)) points spanning x = -3.0 to 3.0 in step increments of 0.5 into the input area.

-3.0 0.0111090
-2.5 0.0439369
-2.0 0.135335
-1.5 0.324652
-1.0 0.606531
-0.5 0.882497
0.0 1.00000
0.5 0.882497
1.0 0.606531
1.5 0.324652
2.0 0.135335
2.5 0.0439369
3.0 0.0111090

Outcome

Returns normalization ⟨ψ|ψ⟩ = 1.772387, ⟨x⟩ ≈ 0, ⟨x²⟩ = 0.499671, and standard deviation σ_x = 0.706874.

2. Displaced Gaussian Wavepacket

Quantum Mechanics Researcher

Background

Testing grid boundary effects on a shifted wavepacket ψ(x) = exp(−(x−1)²/2) sampled across [-3, 3].

Problem

Evaluate the new expectation value ⟨x⟩ and inspect standard deviation changes due to asymmetrical tail truncation.

How to use

Enter 13 discrete coordinates and amplitudes representing the shifted distribution into the tool.

FAQ

What input format does the calculator expect?

Enter whitespace-separated numerical pairs with one `x ψ(x)` per line.

Does the calculator support complex wavefunctions?

No, it evaluates real-valued wavefunctions ψ(x) using the relation |ψ(x)|² = ψ(x)².

Do the x-coordinates need to be sorted before pasting?

No, the tool automatically sorts all distinct x-coordinates in ascending order before integration.

Which numerical integration method is used?

Calculations use the trapezoidal rule across adjacent coordinate intervals.

How is the position uncertainty σ_x calculated?

It is calculated as the standard deviation using the formula σ_x = √(⟨x²⟩ − ⟨x⟩²).

-3.0 0.000335463
-2.5 0.00218749
-2.0 0.0111090
-1.5 0.0439369
-1.0 0.135335
-0.5 0.324652
0.0 0.606531
0.5 0.882497
1.0 1.00000
1.5 0.882497
2.0 0.606531
2.5 0.324652
3.0 0.135335

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Symmetric Gaussian Wavepacket

2. Displaced Gaussian Wavepacket

FAQ

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