Quantum Mechanics — Lecture notes for PHYS223 VIII Mathematical interlude: Hilbert spaces and linear operators X Consequences of the measurement postulate

IX General principles of quantum mechanics

We now have all the tools to formulate the general principles of quantum mechanics, which we state in terms of four postulates.

IX.1 Quantum states

Postulate I: All experimentally accessible information about a quantum system is contained in a state vector $|\psi\rangle$ .

Remarks: An example is the wave function $\psi(x)$ of a point particle in one dimension. In three dimensions a point particle is described by a wave function $\psi(x,y,z)$ . We will also discuss the state of two particles with position ${\bf r_{1}}$ and ${\bf r_{2}}$ , which is described by a wave function $\psi({\bf r_{1}},{\bf r_{2}})$ . Furthermore, we encounter the spin of an electron, which is described by a two-component vector $\psi={\psi_{\uparrow}\choose\psi_{\downarrow}}$ .

IX.2 Observables and operators

Postulate II: Each experimentally observable quantity $\hat{A}$ is represented by a hermitian linear operator.

Remarks: Here, the term observable refers to any measurable quantity. Examples are position $\hat{x}=x$ , momentum $\hat{p}=-i\hbar\frac{d}{dx}$ , energy $\hat{H}=\hat{p}^{2}/2m+V(x)$ , but also kinetic energy $\hat{T}=\hat{p}^{2}/2m$ and potential energy $\hat{V}=V(x)$ can serve as observables. Another example is the angular momentum $\hat{\bf L}=\hat{\bf r}\times\hat{\bf p}$ in three-dimensional systems, where $\hat{\bf r}=(\hat{x},\hat{y},\hat{z})$ is a vector composed of the three position operators, and ${\bf p}=(\hat{p}_{x},\hat{p}_{y},\hat{p}_{z})$ is a vector composed of the three momentum operators $p_{x}=-i\hbar\frac{\partial}{\partial x}$ , $p_{y}=-i\hbar\frac{\partial}{\partial y}$ , $p_{z}=-i\hbar\frac{\partial}{\partial z}$ . Later we will also consider an analogous vector operator $\hat{\bf S}=(\hat{S}_{x},\hat{S}_{y},\hat{S}_{z})$ representing the spin of an electron.

In the context of quantum mechanics, eigenvectors $|a_{n}\rangle$ of an observable $\hat{A}$ are also called eigenstates. The property of hermiticity ensures that all the eigenvalues $a_{n}$ are real, while the eigenstates can be used to construct an orthonormal basis.

IX.3 Dynamics

Postulate III: The time evolution $|\psi(t)\rangle$ of a quantum state is governed by the Schrödinger equation

i\hbar\frac{d}{dt}|\psi(t)\rangle=\hat{H}(t)|\psi(t)\rangle,

(123)

where $\hat{H}$ is a system-specific hermitian operator called Hamiltonian.

Remarks: In the particular case $\hat{H}={\rm const}(t)$ of a time-independent Hamiltonian, the eigenvalues of $\hat{H}$ are denoted as $E_{n}$ , and are interpreted as energies of solutions $|\psi(t)\rangle=\exp(-iE_{n}t/\hbar)|n\rangle$ . Here, the stationary states $|n\rangle$ fulfill the eigenvalue equation $E_{n}|n\rangle=\hat{H}|n\rangle$ , which is known as the stationary Schrödinger equation.

Given an initial state $|\psi(t_{0})\rangle$ , the general solution can always be written as $|\psi(t)\rangle=\hat{U}(t,t_{0})|\psi(t_{0})\rangle$ , where $\hat{U}(t,t_{0})$ is a unitary operator called the time evolution operator. This operator fulfills the Schrödinger equation $i\hbar\frac{d}{dt}\hat{U}(t,t_{0})=\hat{H}(t)\hat{U}(t,t_{0})$ with initial condition $\hat{U}(t_{0},t_{0})=\hat{I}$ . For a time-independent Hamiltonian, the time evolution operator takes the explicit form

\hat{U}(t,t_{0})=\exp[-i(t-t_{0})\hat{H}/\hbar].

(124)

Using the eigenrepresentation $\hat{H}=\sum_{n}E_{n}|n\rangle\langle n|$ of the Hamiltonian we can write $\hat{U}(t,t_{0})=\sum_{n}\exp[-i(t-t_{0})E_{n}/\hbar]|n\rangle\langle n|$ . We will examine time dependence in detail in later chapters.

IX.4 Measurements

The last postulate concerns the remarkable quantum effects which occur when one determines the value of an observable in an experiment. Since this is the most complicated postulate we break it down into three aspects.

IX.4.1 Experimentally observable values

Postulate IVa: In an experiment that determines the value $a$ of an observable with associated operator $\hat{A}$ , the only possible (allowed) observable values are the eigenvalues $a_{n}$ of $\hat{A}$ .

Remarks: E.g., when one measures the energy of a particle bound in a potential well, the only possible results are the discrete energies $E_{n}$ determined by the stationary Schrödinger equation. Because of the hermiticity constraint on such operators, these values are always real.

IX.4.2 Generalised wave function and probability

Postulate IVb: If the normalised state of the system at the time of the measurement is $|\psi\rangle$ , each outcome occurs with probability

P(a)=\sum_{a_{n}=a}|\langle n|\psi\rangle|^{2}=\sum_{a_{n}=a}\langle\psi|\hat{% P}_{n}|\psi\rangle,

(125)

where $|n\rangle$ is the eigenstate associated with $a_{n}$ , and $\hat{P}_{n}=|n\rangle\langle n|$ is the associated projection operator.

Remarks: Using the orthonormality of the eigenstates $|n\rangle$ and the normalisation condition $\langle\psi|\psi\rangle=1$ , it follows that the probability $P(a)$ is automatically normalised: $\sum_{a}P(a)=1$ if $a$ is discrete, and $\int P(a)\,da=1$ if $a$ is continuous.

Disregarding for the moment the possibility that eigenvalues are degenerate, it is useful to interpret the probability in terms of a generalised wave function

\psi_{A}(a_{n})=\langle n|\psi\rangle,

(126)

which consists of the coefficients of the wave function in the eigenbasis of $\hat{A}$ ,

|\psi\rangle=\sum_{n}\psi_{A}(a_{n})|n\rangle.

(127)

The probability $P(a)$ that an experiment returns the value $\hat{A}=a$ is then given by

P(a)=|\psi_{A}(a)|^{2}.

(128)

In one dimension, the generalised wavefunction is given by the overlap integral

\psi_{A}(a_{n})=\int\psi_{n}^{*}(x)\psi(x)\,dx.

(129)

An example is the momentum wave function $\tilde{\psi}(p)$ , discussed in section VII.1, which is calculated with help of the momentum eigenfunction $\psi_{p}(x)=(2\pi\hbar)^{-1/2}\exp(ipx/\hbar)\,dx$ , and corresponds to a Fourier transformation. The factor $(2\pi\hbar)^{-1/2}$ provides that $P(p)=|\tilde{\psi}(p)|^{2}$ is properly normalised.

IX.4.3 Effect of experiments on the system state

Postulate IVc: A measurement with outcome $a$ transforms the quantum state into the state

|\psi^{\prime}\rangle=\sqrt{1/P(a)}\sum_{a_{n}=a}\hat{P}_{n}|\psi\rangle.

(130)

Remarks: Using the decomposition (127) of the pre-measurement state $|\psi\rangle$ in the eigenbasis of the measured observable $\hat{A}$ , the post-measurement state can be written as $|\psi^{\prime}\rangle=\sqrt{1/P(a)}\sum_{a_{n}=a}\psi_{A}(a_{n})|n\rangle$ . Thus, only the components of the eigenstates with eigenvalue compatible to the measured outcome $a$ are retained, and the result is then normalised. It follows that $|\psi^{\prime}\rangle$ is an eigenstate of $\hat{A}$ . In the case that the eigenvalues are not degenerate, a measurement with outcome $a_{n}$ simply transforms the state of the system into the eigenstate $|n\rangle$ .