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Operators and observables

In quantum mechanics, a state contains amplitudes, while an operator is a rule that maps one state to another. An observable—position, momentum, or energy, for example—is represented by a self-adjoint operator. Its possible ideal measurement results are its eigenvalues.

The word “operator” does not automatically mean “measurement.” Translation, time evolution, and projection are also represented by operators. A measurement interpretation requires the relevant observable, its spectral alternatives, and a normalized state.

Linear operators

An operator \(\hat A\) is linear if

\[\hat A(a|\psi\rangle+b|\phi\rangle)=a\hat A|\psi\rangle+b\hat A|\phi\rangle\]

for complex numbers \(a\) and \(b\). In a chosen finite basis, a state is a column vector and the operator is a matrix:

\[|\psi\rangle\longleftrightarrow\begin{pmatrix}c_1\\c_2\\\vdots\end{pmatrix},\qquad \hat A|\psi\rangle\longleftrightarrow A\mathbf c\]

The matrix depends on the basis. The abstract operator and its physical predictions do not. Comparing matrices from different bases entry by entry without transforming them is therefore meaningless.

Complex inner products

For finite vectors, the inner product is

\[\langle\phi|\psi\rangle=\sum_j\phi_j^*\psi_j\]

The left vector is complex-conjugated. Consequently,

\[\langle\psi|\psi\rangle=\sum_j|c_j|^2\geq0\]

and a normalized state has \(\langle\psi|\psi\rangle=1\). Using an ordinary dot product without conjugation can produce a complex or even zero “norm” for a nonzero state.

def complex_inner_product(left, right):
    assert len(left) == len(right)

    total = 0j

    for left_value, right_value in zip(left, right):
        total += left_value.conjugate() * right_value

    return total

For sampled wavefunctions, include quadrature weights. On a uniform one-dimensional grid,

\[\langle\phi|\psi\rangle\approx\sum_j\phi_j^*\psi_j\Delta x\]

The matrix representation of a differential operator and the inner-product weights must be consistent.

Eigenvalues and eigenstates

An eigenstate of \(\hat A\) satisfies

\[\hat A|a_n\rangle=a_n|a_n\rangle\]

The operator changes only the state's scale, so \(a_n\) is the corresponding eigenvalue. If an ideal measurement of \(A\) is made on that eigenstate, the result is \(a_n\) with certainty.

For a complete orthonormal discrete eigenbasis, a normalized state expands as

\[|\psi\rangle=\sum_n c_n|a_n\rangle,\qquad c_n=\langle a_n|\psi\rangle\]

and the Born rule assigns

\[P(a_n)=|c_n|^2=|\langle a_n|\psi\rangle|^2\]

If several independent eigenstates share one eigenvalue, the eigenvalue is degenerate. The probability of that result is the total squared projection into the whole degenerate subspace, not the probability of one arbitrarily selected basis vector.

Hermitian operators

A finite observable matrix must satisfy

\[A^\dagger=A\]

where the dagger means transpose and complex conjugate. Hermitian matrices have real eigenvalues, and eigenvectors belonging to distinct eigenvalues are orthogonal. These properties match the needs of measurement: reported values are real and mutually exclusive alternatives can be represented orthogonally.

Self-adjointness in an infinite-dimensional Hilbert space is more precise than merely writing a formally Hermitian differential expression. The operator's domain and boundary conditions are part of its definition. For example, integration by parts leaves boundary terms; only suitable state domains make the momentum or Hamiltonian genuinely self-adjoint.

import numpy as np

operator = np.array([[1.0, 0.0], [0.0, -1.0]], dtype=complex)
assert np.allclose(operator, operator.conj().T)
print("Hermitian operator:", operator)

Scale the tolerance to the matrix norm. A fixed tiny threshold behaves poorly when matrix elements span very large or very small units.

Expectation value

For a normalized state, the expectation value is

\[\langle A\rangle=\langle\psi|\hat A|\psi\rangle\]

For a state that may not be normalized, use

\[\langle A\rangle=\frac{\langle\psi|\hat A|\psi\rangle}{\langle\psi|\psi\rangle}\]

This is the ensemble mean predicted for many identically prepared measurements. It need not be one of the possible outcomes. If an observable has eigenvalues \(+1\) and \(-1\), an equal superposition can have expectation zero even though no individual ideal measurement returns zero.

def expectation(state, operator):
    norm = np.vdot(state, state).real
    if norm <= 0:
        raise ValueError("State must have positive norm.")
    value = np.vdot(state, operator @ state) / norm
    if abs(value.imag) > 1e-12:
        raise ValueError("Expected a real value for a Hermitian observable.")
    return value.real


state = np.array([np.sqrt(0.75), 0.5j])
print("Expectation:", expectation(state, operator))

A substantial imaginary part for a claimed observable indicates inconsistent complex arithmetic, a non-Hermitian discretization, bad boundaries, or a state outside the operator domain. Do not silently discard it.

Variance and uncertainty

Expectation alone does not describe the spread of outcomes. The variance is

\[(\Delta A)^2=\langle(\hat A-\langle A\rangle)^2\rangle=\langle A^2\rangle-\langle A\rangle^2\]

and the standard deviation is \(\Delta A\). For an exact eigenstate, the variance is zero. A zero expectation does not imply zero uncertainty.

mean = expectation(state, operator)
mean_square = expectation(state, operator @ operator)
variance = max(0.0, mean_square - mean**2)
print("Variance:", variance, "standard deviation:", np.sqrt(variance))

The small clamp handles roundoff only. A significantly negative variance means the implementation or assumptions are wrong.

Position, momentum, and energy

In the position representation, the position operator multiplies by position:

\[\hat x\psi(x)=x\psi(x)\]

The one-dimensional momentum operator is

\[\hat p_x\psi(x)=-i\hbar\frac{d\psi}{dx}\]

and the Hamiltonian for a particle in a potential is

\[\hat H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)\]

Their units differ: position eigenvalues have units of length, momentum eigenvalues have units of momentum, and Hamiltonian eigenvalues have units of energy. A matrix needs its unit and basis metadata; otherwise two arrays with the same dimensions in software may represent physically incompatible operators.

On a finite grid, derivative stencils plus boundary conditions define the momentum and kinetic-energy matrices. A one-sided derivative at one edge can destroy Hermiticity even if every interior stencil looks symmetric.

Commutators and incompatible observables

The commutator is

\[[\hat A,\hat B]=\hat A\hat B-\hat B\hat A\]

Position and momentum obey

\[[\hat x,\hat p_x]=i\hbar\]

which leads to the uncertainty relation

\[\Delta x\,\Delta p_x\geq\frac{\hbar}{2}\]

More generally,

\[\Delta A\,\Delta B\geq\frac12\left|\langle[\hat A,\hat B]\rangle\right|\]

Noncommuting operators generally lack a complete shared eigenbasis. The order of operations can then matter. This is not ordinary measurement noise: it is a structural statement about possible quantum states and outcome distributions.

If two observables commute, they can often be represented by a common eigenbasis, although degeneracies and operator domains require care. Commutation does not mean their numerical matrices are identical.

Basis changes

Let \(U\) be a unitary basis-change matrix. State coefficients and an operator transform as

\[|\psi'\rangle=U|\psi\rangle,\qquad A'=UAU^\dagger\]

The expectation value is unchanged:

\[\langle\psi'|A'|\psi'\rangle=\langle\psi|A|\psi\rangle\]

This makes a strong regression test. Transforming the state but not the operator, or applying the opposite convention to one of them, can return a plausible but basis-dependent result.

Continuous spectra and projectors

Position and free-particle momentum have continuous spectra in ideal unbounded space. Their generalized eigenstates are not normalizable like ordinary finite vectors. Probabilities belong to finite intervals:

\[P(a\leq x\leq b)=\int_a^b|\psi(x)|^2\,dx\]

Projectors provide a common language. If \(\hat P_R\) projects onto outcomes in region \(R\),

\[P(R)=\langle\psi|\hat P_R|\psi\rangle\]

for a normalized state. An ideal yes-or-no projector satisfies

\(\hat P_R^2=\hat P_R\)

and \(\hat P_R^\dagger=\hat P_R\).

Reliable checks

Use independent properties, not one sample output:

Finite matrices make the ideas concrete but do not eliminate domain questions, continuous spectra, degeneracy, or numerical conditioning. This lesson covers ideal projective observables for pure states. Mixed states, generalized measurements, open-system evolution, and detector dynamics require density operators and additional measurement models.

A worked Python notebook

Measure a Hermitian observable whose two eigenvalues are plus one and minus one. The question for this calculation is precise: how does expectation of a plus/minus-one observable change when we change probability of the plus eigenvalue, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

An expectation is an ensemble mean; a single measurement still returns an eigenvalue. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 1 dimensionless. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for probability of the plus eigenvalue, expressed in dimensionless. The output is expectation of a plus/minus-one observable, expressed in dimensionless. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 2 q - 1\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.75 dimensionless, the expression gives 0.5 dimensionless. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("2*q-1", locals={"q": q})
reference_input = 0.75
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "dimensionless")
print(
    "Predicted expectation of a plus/minus-one observable:",
    reference_output,
    "dimensionless",
)

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives -1 dimensionless. The midpoint, at 0.5 dimensionless, gives 0 dimensionless. The upper endpoint gives 1 dimensionless. Read these together with the assumptions: they describe different choices of probability of the plus eigenvalue within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 0.5, 1.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} dimensionless -> {output_value:.6g} dimensionless")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows probability of the plus eigenvalue; the vertical axis shows expectation of a plus/minus-one observable. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 1.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Probability of the plus eigenvalue (dimensionless)")
ax.set_ylabel("Expectation of a plus/minus-one observable (dimensionless)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does expectation of a plus/minus-one observable change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.75 dimensionless is 2 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 2\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.755
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "dimensionless")
print("Full expression:", full_prediction, "dimensionless")
print("Approximation difference:", full_prediction - linear_prediction, "dimensionless")

Here the changed input is 0.755 dimensionless. The tangent estimate is 0.51 dimensionless, while the full expression gives 0.51 dimensionless. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 1.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of probability of the plus eigenvalue inside the stated interval, predict whether expectation of a plus/minus-one observable will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: An expectation is an ensemble mean; a single measurement still returns an eigenvalue. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of dimensionless: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(1.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 1.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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