# Expectation value meaning in quantum mechanics

While reading Griffiths' Introduction to Quantum Mechanics, I met the expected value

$$\langle x \rangle = \int \varPsi^\ast(x )\varPsi (x) \,\mathrm d x$$

and I have got some questions on this. The book says that

$$\langle p \rangle = \int \varPsi^\ast \left(\frac{\hbar} i\frac{\partial}{\partial x} \right) \varPsi (x) \,\mathrm d x$$

and from thease equations comes

$$\langle Q(x,p)\rangle =\int \varPsi^\ast Q \left( x,\frac{\hbar}i\frac{\partial}{\partial x} \right) \varPsi (x) \, \mathrm d x$$

Here $\langle Q(x,p)\rangle$ means "the expectation values of such a quantity", where such refers to the first two equations. But I can't understand, what expectation value is:

1. Is it energy? What will we get if the particle's momentum is $p$ and the position is $x$? Or does it mean something widely different?

2. I couldn't find any information in the book about $Q$ inside the integral. What doest that mean?

• Best to use \langle and \rangle for brackets rather than < and >, they look much nicer. – enumaris May 24 '18 at 20:02
• Note that in quantum mechanics, it is no longer meaningful to say "a particle with momentum $p$ and position $x$." We can only talk about states of a system, and states which correspond to unique values of both $p$ and $x$ do not exist. – J. Murray May 24 '18 at 20:17

The $Q$ here is simply a placeholder for any operator that is a function of $x$ and $p$. For example, if you want to get the expectation of the energy of a harmonic oscillator you would do: $$Q(x,p)\doteq E(x,p)=\frac{p^2}{2m}+\frac{1}{2}kx^2$$ $$\langle E(x,p)\rangle = \frac{1}{2m}\langle p^2\rangle+\frac{k}{2}\langle x^2\rangle = \frac{-\hbar^2}{2m}\int \Psi^*\left( \frac{\partial^2}{\partial^2 x}\right)\Psi dx+\frac{k}{2}\int\Psi^* (x^2)\Psi dx$$
The expectation value of $Q$ is the result of a measurement of $Q$ that QM predicts for a system in the state $\psi$. Common examples of $Q$ are the total energy, the kinetic energy $p^2/2m$, the potential energy $V(x)$ or the angular momentum $\vec r \times \vec p$.
The expectation value of $\langle \psi |\hat Q|\psi\rangle$ is the average result of a measurement of $Q$ for the state $\psi$. If $\psi$ is an eigenstate of $\hat Q$ it is the eigenvalue. This is usually the case if $\hat Q$ is the energy or the angular momentum. If it is not then the expectation value is the sum of (or integral over) the probability of each eigenstate times its eigenvalue. This is usually the case if $\hat Q$ is kinetic or potential energy. Also position or momentum.
It's worth emphasising that the expectation value does not have to be a possible value. The $z$ component of an electron spin can be $+{1 \over 2} \hbar$ or $-{1\over 2} \hbar$. The expectation value for an electron previously polarised along the $x$ direction is $0$ as there is an equal probability of + or -, even though $0$ is not a possible result