Expectation Value of Energy Calculator

Ready to Calculate

Enter a probability model, amplitude list, or Hamiltonian matrix. Results will appear here after submission.

Calculator

Calculation mode

Energy unit

Custom unit label

Output precision

Normalization option

Normalize probability inputs when possible

Supported outputs

Mean energy, second moment, variance, uncertainty, and checks.

Energy levels

Probabilities

Amplitude real parts

Amplitude imaginary parts

Hamiltonian matrix

State vector real parts

State vector imaginary parts

Example Data Table

Case	Energy levels	Probabilities or amplitudes	Expected result
Probability model	0, 1.5, 3.0 eV	0.25, 0.50, 0.25	1.5 eV
Amplitude model	0, 2 eV	Real: 0.6, 0.8. Imaginary: 0, 0	1.28 eV
Matrix model	H = [[1, 0.2], [0.2, 2]]	State real: 0.8, 0.6. Imaginary: 0, 0	1.672 eV

Formula Used

For known probabilities, the expectation value is <E> = Σ pᵢEᵢ.

The second moment is <E²> = Σ pᵢEᵢ².

The variance is σ² = <E²> - <E>².

The standard energy uncertainty is ΔE = √σ².

For amplitudes, pᵢ = |cᵢ|² / Σ|cᵢ|².

For a Hamiltonian matrix, <E> = ψ†Hψ / ψ†ψ.

How to Use This Calculator

Choose a calculation mode from the first field.
Enter values separated by commas, spaces, or new lines.
Use matrix rows on separate lines or separated by semicolons.
Select the unit and precision for the output table.
Enable normalization when values are rounded or unscaled.
Press Calculate and review the result above the form.
Use the CSV or PDF buttons to save your result.

Quantum Energy Expectation Guide

Meaning of the Value

An energy expectation value is the weighted average energy of a quantum state. It is not always one measured result. It is the long run mean predicted by the state and the Hamiltonian. This calculator supports common classroom and lab workflows. You can enter energy levels with probabilities. You can enter complex amplitudes for each level. You can also use a real Hamiltonian matrix with a complex state vector.

Probability Mode

The discrete probability mode is useful when probabilities are already known. The tool multiplies every energy by its probability. Then it adds all products. It also checks the probability sum. Optional normalization can rescale entries when rounding causes a small mismatch.

Amplitude Mode

The amplitude mode is closer to quantum notation. Each coefficient gives a probability through its squared magnitude. The calculator forms each probability from real and imaginary parts. It divides by the total norm when needed. This makes the result match a normalized state. It then calculates the mean, second moment, variance, and uncertainty.

Matrix Mode

The matrix mode evaluates the operator form. It computes the state norm first. Next it calculates the Hamiltonian acting on the state. Then it applies the inner product with the conjugate state. The real part is the expected energy when the Hamiltonian is symmetric. A small imaginary part can warn about entry errors or a non Hermitian operator.

Units and Reports

Use consistent units. Electron volts are convenient for atomic systems. Joules are better for macroscopic energy. The numeric expectation keeps the same unit as the energy inputs. Variance uses squared units. Standard uncertainty returns the original energy unit.

Practical Checks

This page is designed for checking more than one case. The example table shows typical data. The CSV export helps save calculations. The PDF export helps create a clean report. Always compare the normalization value with one. Also inspect variance. A zero variance means the state behaves like a single energy eigenstate.

Advanced Review

Advanced users can test basis changes by comparing entries before and after transformation. The mean energy should stay fixed for an equivalent representation. Different variances can reveal an invalid matrix, mismatched vector order, or copied unit error. Keep labels beside your data. Clear labels make exported files easier to audit later. They also help teachers follow every calculation clearly.

FAQs

What is an energy expectation value?

It is the average energy predicted by a quantum state. It may not equal one measured outcome. It represents the mean result after many identical measurements.

Can I use unnormalized probabilities?

Yes. Keep the normalization option enabled. The calculator divides each probability by the total sum when the sum is positive.

How are amplitudes converted into probabilities?

Each probability comes from the squared magnitude of its complex amplitude. The real and imaginary parts are squared, added, and normalized.

What does the uncertainty output mean?

It is the standard deviation of energy. A larger value means the state has a wider spread across possible energy measurements.

Why can the matrix result show an imaginary part?

A true Hermitian Hamiltonian gives a real expectation value. An imaginary part can appear from nonsymmetric entries, rounding, or incorrect state data.

Which unit should I choose?

Use the same unit as your energy inputs. The result keeps that unit. Variance uses the square of that unit.

Can this handle degenerate energy levels?

Yes. Enter repeated energy values or combine their probabilities. Both methods give the same mean when the total probability is unchanged.

Is this suitable for homework checking?

Yes. It shows the mean, second moment, variance, and normalization checks. You should still show your derivation when required.