1.4 Operators and observables in quantum mechanics

Operators in Quantum Mechanics

Definition and Properties

An operator in quantum mechanics is a mathematical instruction that acts on a function (typically a wavefunction) to produce another function. Each measurable physical quantity, called an observable, has a corresponding operator. Position, momentum, and energy are all examples of observables.

Operators used in quantum mechanics are linear, which means they satisfy two properties:

• Additivity: $\hat{A}(\psi_1 + \psi_2) = \hat{A}\psi_1 + \hat{A}\psi_2$
• Homogeneity: $\hat{A}(c\,\psi) = c\,\hat{A}\psi$, where $c$ is a constant

To extract a measurable prediction from a wavefunction, you calculate the expectation value of an operator. For an operator $\hat{A}$, the expectation value is:

$\langle A \rangle = \int \psi^* \hat{A}\,\psi \, dx$

This integral gives the average result you'd expect if you measured the observable many times on identically prepared systems.

Hermitian Operators and Physical Observables

All operators that represent physical observables must be Hermitian (also called self-adjoint). An operator $\hat{A}$ is Hermitian if it satisfies:

$\int \phi^* (\hat{A}\,\psi)\, dx = \int (\hat{A}\,\phi)^* \psi \, dx$

for all well-behaved functions $\phi$ and $\psi$.

Why does this matter? Hermitian operators guarantee two things that physics demands:

• Real eigenvalues. Measurement outcomes must be real numbers, not complex ones. Hermiticity ensures this.
• Orthonormal eigenfunctions. The eigenfunctions of a Hermitian operator form a complete, orthonormal basis. This means any arbitrary wavefunction can be expanded as a sum of these eigenfunctions, which is the mathematical foundation for calculating measurement probabilities.

The position, momentum, and Hamiltonian operators are all Hermitian.

Common Quantum Operators

Position, Momentum, and Energy Operators

Position operator $\hat{x}$: In the position representation, this operator simply multiplies the wavefunction by $x$:

$\hat{x}\,\psi(x) = x\,\psi(x)$

In three dimensions, the position operator becomes $\hat{\mathbf{r}}$, which multiplies by the position vector.

Momentum operator $\hat{p}$: In the position representation, momentum involves a derivative:

$\hat{p} = -i\hbar \frac{d}{dx}$

The factor of $-i\hbar$ is not arbitrary. It arises from the de Broglie relation connecting wavelength to momentum, and it ensures the operator is Hermitian. In three dimensions, this generalizes to $\hat{\mathbf{p}} = -i\hbar\nabla$.

Hamiltonian operator $\hat{H}$: This represents the total energy of the system. It's built from the kinetic and potential energy operators:

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

The Hamiltonian is central to quantum mechanics because the time-independent Schrödinger equation is just the eigenvalue equation for $\hat{H}$:

$\hat{H}\psi = E\psi$

Angular Momentum Operators

The angular momentum operators $\hat{L}_x$, $\hat{L}_y$, and $\hat{L}_z$ describe rotational motion of a quantum system. They are defined in terms of position and momentum operators. For example:

$\hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x = -i\hbar\frac{\partial}{\partial \phi}$

where $\phi$ is the azimuthal angle in spherical coordinates.

These operators satisfy characteristic commutation relations (covered below) that have deep consequences for how angular momentum is quantized. The operator $\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2$ commutes with each component, which is why you can simultaneously know the total angular momentum and one component (conventionally $\hat{L}_z$), but not two different components at once.

Eigenvalues and Eigenfunctions

Eigenvalue Equation

The eigenvalue equation for an operator $\hat{A}$ is:

$\hat{A}|\psi\rangle = a|\psi\rangle$

Here, $|\psi\rangle$ is an eigenfunction (or eigenstate) of $\hat{A}$, and $a$ is the corresponding eigenvalue. The defining feature is that the operator doesn't change the form of the function; it only scales it by the constant $a$.

The eigenfunctions of a Hermitian operator form a complete, orthonormal basis for the Hilbert space. "Complete" means any wavefunction in that space can be written as a linear combination of these eigenfunctions. "Orthonormal" means distinct eigenfunctions are orthogonal and individually normalized.

Solving the Eigenvalue Equation

Finding eigenvalues and eigenfunctions typically involves these steps:

1. Write out the operator in a specific representation (e.g., position representation).
2. Set up the eigenvalue equation $\hat{A}\psi = a\psi$, which usually becomes a differential equation.
3. Apply boundary conditions appropriate to the physical problem (e.g., the wavefunction must be normalizable, or it must vanish at certain boundaries).
4. Solve for the allowed values of $a$ (the eigenvalues) and the corresponding functions $\psi$ (the eigenfunctions).

For the quantum harmonic oscillator, for instance, solving $\hat{H}\psi_n = E_n\psi_n$ yields the quantized energy levels $E_n = \hbar\omega(n + \frac{1}{2})$ with $n = 0, 1, 2, \ldots$ and eigenfunctions that are Hermite polynomials multiplied by a Gaussian envelope.

In finite-dimensional problems (like spin), the eigenvalue equation becomes a matrix equation, and you find eigenvalues by solving $\det(\mathbf{A} - a\mathbf{I}) = 0$.

Significance of Eigenvalues and Eigenfunctions

Eigenvalues of an operator are the only possible outcomes of measuring that observable. You will never measure a value that isn't an eigenvalue of the corresponding operator.

If a system is already in an eigenstate $|\psi_n\rangle$ of $\hat{A}$, measuring $A$ will yield the eigenvalue $a_n$ with certainty.

If the system is in a general state $|\Psi\rangle$, you expand it in the eigenbasis:

$|\Psi\rangle = \sum_n c_n |\psi_n\rangle$

The probability of measuring eigenvalue $a_n$ is $|c_n|^2$. This is the Born rule, and it connects the mathematical formalism of operators directly to experimental predictions.

Commutation Relations and Observables

Commutation Relations

The commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as:

$[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$

Whether two operators commute determines whether you can know both observables precisely at the same time.

• If $[\hat{A}, \hat{B}] = 0$ (they commute): the two observables can be measured simultaneously with arbitrary precision. The system can exist in a simultaneous eigenstate of both operators, and measuring one doesn't disturb the value of the other.
• If $[\hat{A}, \hat{B}] \neq 0$ (they don't commute): the two observables cannot both be precisely determined at the same time. Measuring one necessarily introduces uncertainty into the other.

Heisenberg Uncertainty Principle

The most important example of non-commuting operators is position and momentum. Their commutator is:

$[\hat{x}, \hat{p}] = i\hbar$

You can verify this by acting on a test function $f(x)$:

1. Compute $\hat{x}\hat{p}f = x\left(-i\hbar \frac{df}{dx}\right)$
2. Compute $\hat{p}\hat{x}f = -i\hbar \frac{d}{dx}(xf) = -i\hbar\left(f + x\frac{df}{dx}\right)$
3. Subtract: $[\hat{x}, \hat{p}]f = i\hbar\, f$

Since this holds for any $f$, the commutator is $i\hbar$.

This non-zero commutator leads directly to the Heisenberg uncertainty principle:

$\Delta x \, \Delta p \geq \frac{\hbar}{2}$

where $\Delta x$ and $\Delta p$ are the standard deviations (uncertainties) in position and momentum. No matter how clever your experiment, you cannot beat this bound. It's not a limitation of measurement technology; it's a fundamental property of nature encoded in the operator algebra.

More generally, for any two observables with operators $\hat{A}$ and $\hat{B}$:

$\Delta A \, \Delta B \geq \frac{1}{2}|\langle[\hat{A}, \hat{B}]\rangle|$

Role in Quantum Theory

Commutation relations are structural pillars of quantum mechanics. The canonical commutation relation $[\hat{x}, \hat{p}] = i\hbar$ is sometimes taken as a starting axiom of the theory rather than a derived result.

These relations are also essential for:

• Angular momentum algebra: The commutation relations $[\hat{L}_i, \hat{L}_j] = i\hbar\epsilon_{ijk}\hat{L}_k$ determine the quantization of angular momentum and the allowed quantum numbers $l$ and $m_l$.
• Ladder operator methods: Commutation relations let you construct raising and lowering operators (e.g., for the harmonic oscillator or angular momentum), which provide elegant algebraic solutions without solving differential equations directly.
• Quantum field theory: The extension of commutation relations to field operators underlies the entire framework of second quantization.

Understanding how operators commute (or don't) is what separates quantum mechanics from classical mechanics, where all observables can in principle be known simultaneously.