KAHIBARO
Table of Contents
From physical quantities to quantum operators
In classical physics, a measurable quantity such as position, momentum, or energy is treated as an ordinary number once the state of the system is known. In quantum mechanics, the situation is different. The state of a system is described by a wave function, and measurable quantities are represented by mathematical objects called operators.
An operator is a rule that acts on a wave function and produces another function. Observables are the physical quantities that can be measured in an experiment. In quantum mechanics, each observable is associated with an operator.
For example, if the wave function is $\psi(x)$, then the position operator and momentum operator in one dimension are
$$
\hat{x}\psi(x) = x\psi(x)
$$
and
$$
\hat{p}\psi(x) = -i\hbar \frac{d}{dx}\psi(x).
$$
The hat on top of a symbol reminds us that it is an operator, not an ordinary number.
In quantum mechanics, physical observables are represented by operators acting on the wave function.
How operators act
Some operators are simple multiplication rules. The position operator multiplies the wave function by the coordinate $x$. Other operators involve differentiation. The momentum operator contains a derivative, so it depends on how the wave function changes with position.
This means operators are not just labels. They actively transform the wave function. If two different operators act on the same wave function, the results are usually different.
Consider a wave function $\psi(x)$. Then
$$
\hat{x}\psi(x) = x\psi(x)
$$
while
$$
\hat{p}\psi(x) = -i\hbar \frac{d\psi}{dx}.
$$
These two results are generally not the same kind of function.
Common operators in one dimension
Some of the most important quantum operators are shown below.
| Observable | Operator |
|---|---|
| Position | $\hat{x} = x$ |
| Momentum | $\hat{p} = -i\hbar \dfrac{d}{dx}$ |
| Kinetic energy | $\hat{T} = \dfrac{\hat{p}^2}{2m} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}$ |
| Total energy, Hamiltonian | $\hat{H} = \hat{T} + V(x)$ |
If the potential energy is $V(x)$, then the Hamiltonian operator in one dimension is
$$
\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x).
$$
The Hamiltonian is especially important because it represents the total energy of the system.
The Hamiltonian operator is the energy operator. In one dimension,
$$
\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x).
$$
Eigenvalues and eigenfunctions
A central idea in quantum mechanics is that a measurement of an observable gives one of the allowed values of its operator. These allowed values are called eigenvalues.
If an operator $\hat{A}$ acts on a wave function $\psi$ and produces the same function multiplied by a number $a$, then
$$
\hat{A}\psi = a\psi.
$$
In this case, $\psi$ is an eigenfunction of $\hat{A}$, and $a$ is the corresponding eigenvalue.
This equation means that the state $\psi$ has a definite value of the observable associated with $\hat{A}$. If the system is in such a state, measuring that observable will give the value $a$.
For example, if
$$
\hat{H}\psi = E\psi,
$$
then $\psi$ is an energy eigenfunction and $E$ is an allowed energy value.
If
$$
\hat{A}\psi = a\psi,
$$
then $\psi$ is an eigenfunction of $\hat{A}$ and $a$ is the measurable value, called the eigenvalue.
A simple example with momentum
Take the wave function
$$
\psi(x) = Ae^{ikx},
$$
where $A$ and $k$ are constants. Apply the momentum operator:
$$
\hat{p}\psi(x) = -i\hbar \frac{d}{dx}\left(Ae^{ikx}\right).
$$
Since
$$
\frac{d}{dx}e^{ikx} = ike^{ikx},
$$
we get
$$
\hat{p}\psi(x) = -i\hbar (ik)Ae^{ikx} = \hbar k \psi(x).
$$
So this wave function is an eigenfunction of the momentum operator, with eigenvalue
$$
p = \hbar k.
$$
This means the state has a definite momentum.
Commutators and measurement order
In classical physics, multiplying quantities usually does not depend on order. In quantum mechanics, the order of operators can matter.
For two operators $\hat{A}$ and $\hat{B}$, the commutator is defined as
$$
[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}.
$$
If the commutator is zero, the operators commute:
$$
[\hat{A},\hat{B}] = 0.
$$
If they do not commute, then applying them in different orders gives different results.
The most famous example is position and momentum:
$$
[\hat{x},\hat{p}] = i\hbar.
$$
This nonzero result is deeply important. It tells us that position and momentum are not simultaneously definite in the same way as in classical physics.
Position and momentum do not commute:
$$
[\hat{x},\hat{p}] = i\hbar.
$$
Noncommuting operators are a key feature of quantum mechanics.
Why commuting operators matter
If two observables are represented by commuting operators, they can have common eigenfunctions. In that case, the system can have definite values of both observables at the same time.
If two operators do not commute, then in general they cannot both have sharply defined values in the same state.
This idea connects directly to uncertainty, but the full uncertainty principle belongs to a separate chapter. Here the main point is mathematical and physical: commutation tells us whether two measurements are compatible.
| Operator relation | Physical meaning |
|---|---|
| $[\hat{A},\hat{B}] = 0$ | Observables are compatible |
| $[\hat{A},\hat{B}] \neq 0$ | Observables are generally incompatible |
Hermitian operators
A physical observable must always produce real measurement values, not imaginary ones. For this reason, observable operators in quantum mechanics are Hermitian operators.
A Hermitian operator has the property that its eigenvalues are real, and eigenfunctions belonging to different eigenvalues are orthogonal. These mathematical properties make Hermitian operators suitable for representing measurable quantities.
You do not need the full abstract definition yet, but the key physical consequence is simple.
Observable operators must be Hermitian, because measured values must be real numbers.
This is why operators like $\hat{x}$, $\hat{p}$, and $\hat{H}$ are constructed carefully so that they represent physical measurements correctly.
Expectation values as average measurement results
Even when a system is not in an eigenstate of an observable, we can still predict the average result of many repeated measurements. This average is called the expectation value.
For an operator $\hat{A}$ and a normalized wave function $\psi(x)$, the expectation value is
$$
\langle A \rangle = \int_{-\infty}^{\infty} \psi^*(x)\,\hat{A}\,\psi(x)\,dx.
$$
Here $\psi^*(x)$ is the complex conjugate of $\psi(x)$.
For position,
$$
\langle x \rangle = \int_{-\infty}^{\infty} \psi^*(x)\,x\,\psi(x)\,dx.
$$
For momentum,
$$
\langle p \rangle = \int_{-\infty}^{\infty} \psi^*(x)\left(-i\hbar \frac{d}{dx}\right)\psi(x)\,dx.
$$
This formula does not usually give the result of one single measurement. Instead, it gives the average value obtained from many identical experiments.
Operators in different representations
So far, we have used the position representation, where the wave function depends on $x$. In that representation, the position operator is simple multiplication, while the momentum operator is a derivative.
In other representations, the same observable may have a different mathematical form. The physical quantity is the same, but the operator looks different depending on how the state is expressed.
This shows that the operator is tied to the observable, while its exact appearance depends on the mathematical representation.
Visual picture of operator action
The figure below shows the idea that operators act on a wave function to produce another function.
The operator changes the wave function according to a mathematical rule. If the new function is just a constant times the original function, then the original function is an eigenfunction of that operator.
Summary of the main ideas
Operators are the quantum mechanical representation of measurable quantities. Observables such as position, momentum, and energy correspond to specific operators. These operators act on wave functions, often by multiplication or differentiation. If a wave function satisfies an eigenvalue equation, then it represents a state with a definite value of that observable. The order of operators can matter, and this is expressed through commutators. Physical observables must be represented by Hermitian operators so that measurement results are real. Finally, expectation values give the average result of many measurements.
Key formulas of this chapter:
$$
\hat{x} = x
$$
$$
\hat{p} = -i\hbar \frac{d}{dx}
$$
$$
\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)
$$
$$
\hat{A}\psi = a\psi
$$
$$
[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}
$$
$$
[\hat{x},\hat{p}] = i\hbar
$$
$$
\langle A \rangle = \int \psi^* \hat{A}\psi \, dx
$$