16.3 Operators and observables

Operators in Quantum Mechanics

Definition and Role of Operators

In quantum mechanics, an operator is a mathematical instruction that acts on a function and produces another function. Operators represent physical quantities you can measure, like position, momentum, and energy. When an operator acts on a wave function, it extracts information about that physical quantity from the quantum state.

A few key properties to know:

• Linear operators satisfy $\hat{A}(c_1\psi_1 + c_2\psi_2) = c_1\hat{A}\psi_1 + c_2\hat{A}\psi_2$. Almost all operators you'll encounter in this course are linear.
• Hermitian (self-adjoint) operators are the ones that correspond to real, measurable quantities. Their defining property is that they always produce real eigenvalues, which makes physical sense since measurement outcomes are real numbers.

Here are the most common operators you need to know:

Eigenvalues and Eigenfunctions

An eigenvalue equation has the form:

$\hat{A}\phi_i = a_i\phi_i$

Here, $\hat{A}$ is the operator, $\phi_i$ is an eigenfunction, and $a_i$ is the corresponding eigenvalue. The operator acts on the eigenfunction and returns the same function, just scaled by a constant. That constant is the eigenvalue.

Why does this matter physically?

• Eigenvalues are the only possible outcomes you can get when you measure the associated observable. If you measure energy, you can only ever get one of the eigenvalues of $\hat{H}$.
• Eigenfunctions are the states the system is in when a measurement yields that specific eigenvalue. After measuring energy and getting $E_n$, the system is in the eigenfunction $\phi_n$.

For example, the eigenvalues of the Hamiltonian operator correspond to the allowed energy levels of the system, and the eigenfunctions are the stationary states.

Operators vs. Observables

Relationship between Operators and Observables

An observable is any physical quantity you can measure: position, momentum, energy, angular momentum. In quantum mechanics, every observable has a corresponding Hermitian operator. The connection works like this:

• The operator encodes the mathematical recipe for extracting information about the observable from a wave function.
• The eigenvalues of that operator are the possible measurement results.
• The eigenfunctions are the states associated with those results.

This is a core idea in the theory. You never calculate an observable directly from a wave function; you always go through the operator.

Calculating Expectation Values

The expectation value of an observable gives the average result you'd get if you measured that quantity on many identical copies of the system. It's calculated as:

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

For position:

$\langle x \rangle = \int \psi^*(x)\, x\, \psi(x)\, dx$

For momentum:

$\langle p \rangle = \int \psi^*(x)\left(-i\hbar \frac{d}{dx}\right)\psi(x)\, dx$

Note that the expectation value doesn't have to equal any particular eigenvalue. It's a weighted average over all possible outcomes. If the system is in an eigenstate of $\hat{A}$, then the expectation value equals that eigenvalue exactly, with zero uncertainty.

Applying Quantum Postulates

Postulates of Quantum Mechanics

The postulates provide the rules connecting the math to physical predictions. For this topic, four are directly relevant:

1. State postulate: The state of a quantum system is fully described by a wave function $\psi$ (or state vector $|\psi\rangle$) in a Hilbert space.
2. Observable postulate: Every measurable quantity is represented by a Hermitian operator. The only possible measurement outcomes are the eigenvalues of that operator.
3. Probability postulate: If the system is in state $|\psi\rangle$, the probability of measuring eigenvalue $a_i$ is $P(a_i) = |\langle \phi_i | \psi \rangle|^2$, where $|\phi_i\rangle$ is the eigenfunction for $a_i$.
4. Time evolution postulate: The state evolves according to the time-dependent Schrödinger equation: $i\hbar \frac{\partial}{\partial t}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle$.

Solving Problems with Operators and Observables

Here's how these postulates come together in a typical problem:

To find the probability of a specific measurement outcome $a_i$:

1. Identify the operator $\hat{A}$ for the observable.
2. Find its eigenfunctions $|\phi_i\rangle$ and eigenvalues $a_i$.
3. Compute the overlap $c_i = \langle \phi_i | \psi \rangle$.
4. The probability is $P(a_i) = |c_i|^2$.

To find the time evolution of a state:

1. Expand $|\psi(0)\rangle$ in the energy eigenbasis: $|\psi(0)\rangle = \sum_n c_n |\phi_n\rangle$.
2. Attach the time-dependent phase to each term: $|\psi(t)\rangle = \sum_n c_n\, e^{-iE_nt/\hbar} |\phi_n\rangle$.

This works because energy eigenstates have a particularly simple time dependence.

Eigenvalues and Eigenfunctions

Completeness and Expansion of Quantum States

The eigenfunctions of a Hermitian operator form a complete set. This means any valid quantum state can be written as a linear combination of those eigenfunctions:

$|\psi\rangle = \sum_i c_i |\phi_i\rangle$

The expansion coefficients are found by projection:

$c_i = \langle \phi_i | \psi \rangle$

Each coefficient $c_i$ carries physical meaning. Its squared modulus $|c_i|^2$ gives the probability of measuring the eigenvalue $a_i$. Since probabilities must add to 1, you get the normalization condition:

$\sum_i |c_i|^2 = 1$

This expansion is one of the most useful tools in quantum mechanics. It converts an abstract state into a concrete list of "how much" of each eigenstate is present, and from there you can calculate any expectation value or probability you need.

Commutation Relations

Definition and Properties

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

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

If $[\hat{A}, \hat{B}] = 0$, the operators commute, meaning the order you apply them doesn't matter. If the commutator is nonzero, the order matters and the operators are non-commuting.

Why you should care:

• Commuting operators share a common set of eigenfunctions. You can know both observables simultaneously with perfect precision.
• Non-commuting operators cannot share a complete set of eigenfunctions. There's a fundamental limit on how precisely you can know both quantities at once.

Implications: The Uncertainty Principle

The most important commutation relation in quantum mechanics is between position and momentum:

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

This nonzero commutator leads directly to the Heisenberg uncertainty principle:

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

This isn't a statement about measurement equipment being imprecise. It's a fundamental property of nature: a quantum state cannot simultaneously have a perfectly defined position and a perfectly defined momentum.

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

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

Another example: the angular momentum components $\hat{L}_x$, $\hat{L}_y$, and $\hat{L}_z$ don't commute with each other (e.g., $[\hat{L}_x, \hat{L}_y] = i\hbar\hat{L}_z$). This means you can't simultaneously know all three components of angular momentum. However, each component commutes with $\hat{L}^2$, so you can know the total angular momentum magnitude and one component at the same time.