---
title: "Operators and expectation values of observables"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:operators-expectation-values"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/operators-expectation-values
site: BCT Engineering
---

# Operators and expectation values of observables

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2070 Asar, 8 marks
- Video: [Expectation Values ||Quantum mechanics Lec 25 #csirnet #quantumphysics #gate #physicalscience](https://www.youtube.com/watch?v=8YDGHY1p3JA) — Target Physics Exams 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/operators-expectation-values
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

What are the operators in quantum mechanics? Explain their uses in deducing the expected values of observable quantity.

## Short answer

In quantum mechanics, operators are mathematical instructions corresponding to observables (e.g., position, momentum, energy). They act on wavefunctions to yield eigenvalues (definite measured values) or expectation values via ⟨A⟩ = ∫ ψ* Â ψ dx, giving average outcomes and enabling uncertainty calculations.

## Answer

In quantum mechanics an **operator** is a mathematical instruction that acts on a wave function $\psi$ to give another function. Every measurable (observable) quantity such as position, momentum or energy has a corresponding operator. When the operator acts on $\psi$ and returns a constant times $\psi$, that constant is the value we would measure.

#### Common operators

| Observable | Classical form | Operator |
|---|---|---|
| Position | $x$ | $\hat{x} = x$ |
| Momentum | $p_x$ | $\hat{p}_x = -i\hbar \dfrac{\partial}{\partial x}$ |
| Kinetic energy | $\dfrac{p^2}{2m}$ | $-\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2}$ |
| Potential energy | $V(x)$ | $V(x)$ |
| Total energy (Hamiltonian) | $\dfrac{p^2}{2m} + V$ | $\hat{H} = -\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2} + V(x)$ |
| Energy (time form) | $E$ | $\hat{E} = i\hbar \dfrac{\partial}{\partial t}$ |

Writing $\hat{H}\psi = \hat{E}\psi$ gives the Schrödinger equation itself, so the operators are the basis of quantum mechanics.

#### Eigenvalue equation

If $\hat{A}\psi = a\psi$ with $a$ a constant, then $\psi$ is an **eigenfunction** of $\hat{A}$ and $a$ is its **eigenvalue**. A measurement of $A$ on this state always gives exactly $a$. Example: the time-independent Schrödinger equation $\hat{H}\psi = E\psi$ gives the allowed energies $E$.

#### Expectation value

In general a state is not an eigenfunction, so repeated measurements give different results. The **expectation value** is the average of many measurements on identical systems. Since $|\psi|^2 dx$ is the probability of finding the particle between $x$ and $x+dx$:

$$
\langle A \rangle = \frac{\int_{-\infty}^{\infty} \psi^*\, \hat{A}\, \psi \, dx}{\int_{-\infty}^{\infty} \psi^* \psi \, dx}
$$

For a normalised wave function the denominator is 1. Then:

$$
\langle x \rangle = \int \psi^* x\, \psi\, dx, \qquad \langle p \rangle = \int \psi^* \left(-i\hbar \frac{\partial \psi}{\partial x}\right) dx, \qquad \langle E \rangle = \int \psi^* \hat{H} \psi\, dx
$$

The operator is placed between $\psi^*$ and $\psi$ so that derivatives act only on $\psi$.

#### Example: particle in an infinite well

For $\psi_n = \sqrt{2/L}\,\sin(n\pi x/L)$ in $0 \le x \le L$:

- $\langle x \rangle = \frac{2}{L}\int_0^L x \sin^2(n\pi x/L)\, dx = \frac{L}{2}$, the centre of the well, as symmetry suggests.
- $\langle p \rangle = 0$, because the particle moves left and right with equal probability.
- $\hat{H}\psi_n = \frac{n^2h^2}{8mL^2}\psi_n$, so $\langle E \rangle = E_n$ exactly: $\psi_n$ is an energy eigenfunction.

#### Uses

1. They give the possible measured values (eigenvalues), e.g. quantised energy levels of electrons in atoms and solids.
2. They give average values of position, momentum and energy for any state through $\langle A \rangle$.
3. They allow uncertainties, $\Delta x = \sqrt{\langle x^2\rangle - \langle x\rangle^2}$, to be calculated, linking to the uncertainty principle.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the momentum operator in 1-D?
   A. p = x
   B. -i hbar d/dx
   C. i hbar d/dt
   D. -hbar^2/2m d^2/dx^2
   Answer: B. p_x operator = -i hbar partial/partial x. (Key point: Momentum operator)

2. If A psi = a psi with constant a, what does a represent?
   A. The probability density
   B. The value always obtained when A is measured in that state
   C. The average over many different states
   D. The wave function norm
   Answer: B. psi is an eigenfunction and a its eigenvalue, the definite measurement result. (Key point: Eigenvalue equation)

3. Which is the expectation value of A for a normalised psi?
   A. Integral of psi A psi* dx taken with A acting on psi*
   B. Integral of psi* A psi dx
   C. Integral of A |psi|^2 inside the operator only
   D. Integral of psi* psi dx
   Answer: B. <A> = integral psi* (A psi) dx; the operator acts on psi so derivatives affect only psi. (Key point: Expectation value)

4. For the infinite well state psi_n, what is <p>?
   A. n h / L
   B. hbar k
   C. 0
   D. L/2
   Answer: C. Particle moves left and right equally, so <p> = 0; <x> = L/2. (Key point: Well example)

## Related in Electrical Engineering Material

- [Operators and expected energy in a well](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aoperators-expected-energy-well) — Electrical Engineering Material, chapter 1
- [Classical versus quantum mechanics](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aclassical-vs-quantum-mechanics) — Electrical Engineering Material, chapter 1, asked 2×
- [Quantum mechanics and golf ball wavelength](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Agolf-ball-de-broglie-wavelength) — Electrical Engineering Material, chapter 1

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