Comprehensive Study Notes

This section summarizes the postulates of quantum mechanics introduced in previous chapters.

POSTULATE 1. The state of a system is described by a function $\Psi$ of the coordinates of the particles and the time. This function, called the state function or wave function, contains all the information that can be determined about the system. We further postulate that $\Psi$ is single-valued, continuous, and quadratically integrable. For continuum states, the quadratic integrability requirement is omitted.

The designation "wave function" for $\Psi$ is perhaps not the best choice. A physical wave moving in three-dimensional space is a function of the three spatial coordinates and the time. However, for an $n$-particle system, the function $\Psi$ is a function of $3 n$ spatial coordinates and the time. Hence, for a many-particle system, we cannot interpret $\Psi$ as any sort of physical wave. The state function is best thought of as a function from which we can calculate various properties of the system. The nature of the information that $\Psi$ contains is the subject of Postulate 5 and its consequences.

POSTULATE 2. To every physically observable property there corresponds a linear Hermitian operator. To find this operator, write down the classical-mechanical expression for the observable in terms of Cartesian coordinates and corresponding linearmomentum components, and then replace each coordinate $x$ by the operator $x$. and each momentum component $p_{x}$ by the operator $-i \hbar \partial / \partial x$.

We saw in Section 7.2 that the restriction to Hermitian operators arises from the requirement that average values of physical quantities be real numbers. The requirement of linearity is closely connected to the superposition of states discussed in Section 7.6. In our derivation of (7.70) for the average value of a property $B$ for a state that was expanded as a superposition of the eigenfunctions of $\hat{B}$, the linearity of $\hat{B}$ played a key role.

When the classical quantity contains a product of a Cartesian coordinate and its conjugate momentum, we run into the problem of noncommutativity in constructing the correct quantum-mechanical operator. Several different rules have been proposed to handle this case. See J. R. Shewell, Am. J. Phys., 27, 16 (1959); E. H. Kerner and W. G. Sutcliffe, J. Math. Phys., 11, 391 (1970); A. de Souza Dutra, J. Phys. A: Math. Gen., 39, 203 (2006) (arxiv.org/abs/0705.3247).

The process of finding quantum-mechanical operators in non-Cartesian coordinates is complicated. See K. Simon, Am. J. Phys., 33, 60 (1965); G. R. Gruber, Found. Phys., 1, 227 (1971).

POSTULATE 3. The only possible values that can result from measurements of the physically observable property $B$ are the eigenvalues $b{i}$ in the equation $\hat{B} g{i}=b{i} g{i}$, where $\hat{B}$ is the operator corresponding to the property $B$. The eigenfunctions $g_{i}$ are required to be well behaved.

Our main concern is with the energy levels of atoms and molecules. These are given by the eigenvalues of the energy operator, the Hamiltonian $\hat{H}$. The eigenvalue equation for $\hat{H}, \hat{H} \psi=E \psi$, is the time-independent Schrödinger equation. However, finding the possible values of any property involves solving an eigenvalue equation.

POSTULATE 4. If $\hat{B}$ is a linear Hermitian operator that represents a physically observable property, then the eigenfunctions $g_{i}$ of $\hat{B}$ form a complete set.

There are Hermitian operators whose eigenfunctions do not form a complete set (see Griffiths, pp. 99, 106; Messiah, p. 188; Ballentine, Sec. 1.3). The completeness requirement is essential to developing the theory of quantum mechanics, so it is necessary to postulate that all Hermitian operators that correspond to observable properties have a complete set of eigenfunctions. Postulate 4 allows us to expand the wave function for any state as a superposition of the orthonormal eigenfunctions of any quantum-mechanical operator:

\(
\begin{equation}
\Psi=\sum{i} c{i} g{i}=\sum{i}\left|g{i}\right\rangle\left\langle g{i} \mid \Psi\right\rangle \tag{7.95}
\end{equation}
\)

POSTULATE 5. If $\Psi(q, t)$ is the normalized state function of a system at time $t$, then the average value of a physical observable $B$ at time $t$ is

\(
\begin{equation}
\langle B\rangle=\int \Psi^{} \hat{B} \Psi d \tau \tag{7.96}
\end{}
\)

The definition of the quantum-mechanical average value is given in Section 3.7 and should not be confused with the time average used in classical mechanics.

From Postulates 4 and 5, we showed in Section 7.6 that the probability of observing the nondegenerate eigenvalue $b{i}$ in a measurement of $B$ is $P\left(b{i}\right)=\left|\int g{i}^{*} \Psi d \tau\right|^{2}=$ $\left|\left\langle g{i} \mid \Psi\right\rangle\right|^{2}$, where $\hat{B} g{i}=b{i} g{i}$. If $\Psi$ happens to be one of the eigenfunctions of $\hat{B}$, that is, if $\Psi=g{k}$, then $P\left(b{i}\right)$ becomes $P\left(b{i}\right)=\left|\int g{i}^{*} g{k} d \tau\right|^{2}=\left|\delta{i k}\right|^{2}=\delta{i k}$, where the orthonormality of the eigenfunctions of the Hermitian operator $\hat{B}$ was used. We are certain to observe the value $b{k}$ when $\Psi=g{k}$.

POSTULATE 6. The time development of the state of an undisturbed quantum-mechanical system is given by the Schrödinger time-dependent equation

\(
\begin{equation}
-\frac{\hbar}{i} \frac{\partial \Psi}{\partial t}=\hat{H} \Psi \tag{7.97}
\end{equation}
\)

where $\hat{H}$ is the Hamiltonian (that is, energy) operator of the system.

The time-dependent Schrödinger equation is a first-order differential equation in the time, so that, just as in classical mechanics, the present state of an undisturbed system
determines the future state. However, unlike knowledge of the state in classical mechanics, knowledge of the state in quantum mechanics involves knowledge of only the probabilities for various possible outcomes of a measurement. Thus, suppose we have several identical noninteracting systems, each having the same state function $\Psi\left(t{0}\right)$ at time $t{0}$. If we leave each system undisturbed, then the state function for each system will change in accord with (7.97). Since each system has the same Hamiltonian, each system will have the same state function $\Psi\left(t{1}\right)$ at any future time $t{1}$. However, suppose that at time $t{2}$ we measure property $B$ in each system. Although each system has the same state function $\Psi\left(t{2}\right)$ at the instant the measurement begins, we will not get the same result for each system. Rather, we will get a spread of possible values $b{i}$, where $b{i}$ are the eigenvalues of $\hat{B}$. The relative number of times we get each $b{i}$ can be calculated from the quantities $\left|c{i}\right|^{2}$, where $\Psi\left(t{2}\right)=\sum{i} c{i} g{i}$ with the $g_{i}$ 's being the eigenfunctions of $\hat{B}$.

If the Hamiltonian is independent of time, we have the possibility of states of definite energy $E$. For such states the state function must satisfy

\(
\begin{equation}
\hat{H} \Psi=E \Psi \tag{7.98}
\end{equation}
\)

and the time-dependent Schrödinger equation becomes

\(
-\frac{\hbar}{i} \frac{\partial \Psi}{\partial t}=E \Psi
\)

which integrates to $\Psi=A e^{-i E t / \hbar}$, where $A$, the integration "constant," is independent of time. The function $\Psi$ depends on the coordinates and the time, so $A$ is some function of the coordinates, which we designate as $\psi(q)$. We have

\(
\begin{equation}
\Psi(q, t)=e^{-i E t / \hbar} \psi(q) \tag{7.99}
\end{equation}
\)

for a state of constant energy. The function $\psi(q)$ satisfies the time-independent Schrödinger equation

\(
\hat{H} \psi(q)=E \psi(q)
\)

which follows from (7.98) and (7.99). The factor $e^{-i E t / \hbar}$ simply indicates a change in the phase of the wave function $\Psi(q, t)$ with time and has no direct physical significance. Hence we generally refer to $\psi(q)$ as "the wave function." The Hamiltonian operator plays a unique role in quantum mechanics in that it occurs in the fundamental dynamical equation, the time-dependent Schrödinger equation. The eigenstates of $\hat{H}$ (known as stationary states) have the special property that the probability density $|\Psi|^{2}$ is independent of time.

The time-dependent Schrödinger equation (7.97) is $(i \hbar \partial / \partial t-\hat{H}) \Psi=0$. Because the operator $i \hbar \partial / \partial t-\hat{H}$ is linear, any linear combination of solutions of the time-dependent Schrödinger equation (7.97) is a solution of (7.97). For example, if the Hamiltonian $\hat{H}$ is independent of time, then there exist stationary-state solutions $\Psi{n}=e^{-i E{n} t / \hbar} \psi_{n}(q)$ [Eq. (7.99)] of the time-dependent Schrödinger equation. Any linear combination

\(
\begin{equation}
\Psi=\sum{n} c{n} \Psi{n}=\sum{n} c{n} e^{-i E{n} t \hbar} \psi_{n}(q) \tag{7.100}
\end{equation}
\)

where the $c{n}$ 's are time-independent constants is a solution of the time-dependent Schrödinger equation, although it is not an eigenfunction of $\hat{H}$. Because of the completeness of the eigenfunctions $\psi{n}$, any state function can be written in the form (7.100) if $\hat{H}$ is independent of time. (See also Section 9.8.) The state function (7.100) represents a state that does not have a definite energy. Rather, when we measure the energy, the probability of getting $E{n}$ is $\left|c{n} e^{-i E{n} t / \hbar}\right|^{2}=\left|c{n}\right|^{2}$.

To find the constants $c{n}$ in (7.100), we write (7.100) at time $t{0}$ as $\Psi\left(q, t{0}\right)=$ $\sum{n} c{n} e^{-i E{n} t{0} / \hbar} \psi{n}(q)$. Multiplication by $\psi_{j}^{*}(q)$ followed by integration over all space gives

\(
\left\langle\psi{j}(q) \mid \Psi\left(q, t{0}\right)\right\rangle=\sum{n} c{n} e^{-i E{n} t{0} / \hbar}\left\langle\psi{j} \mid \psi{n}\right\rangle=\sum{n} c{n} e^{-i E{n} t{0} / \hbar} \delta{j n}=c{j} e^{-i E{j} t{0} / \hbar}
\)

so $c{j}=\left\langle\psi{j} \mid \Psi\left(q, t{0}\right)\right\rangle e^{i E{j} t_{0} / \hbar}$ and (7.100) becomes

\(
\begin{equation}
\Psi(q, t)=\sum{j}\left\langle\psi{j}(q) \mid \Psi\left(q, t{0}\right)\right\rangle e^{-i E{j}\left(t-t{0}\right) / \hbar} \psi{j}(q) \quad \text { if } \hat{H} \text { ind. of } t \tag{7.101}
\end{equation}
\)

where $\hat{H} \psi{j}=E{j} \psi{j}$ and the $\psi{j}$ 's have been chosen to be orthonormal. Equation (7.101) tells us how to find $\Psi$ at time $t$ from $\Psi$ at an initial time $t_{0}$ and is the general solution of the time-dependent Schrödinger equation when $\hat{H}$ is independent of $t$. [Equation (7.101) can also be derived directly from the time-dependent Schrödinger equation; see Prob. 7.46.]

EXAMPLE

A particle in a one-dimensional box of length $l$ has a time-independent Hamiltonian and has the state function $\Psi=2^{-1 / 2} \psi{1}+2^{-1 / 2} \psi{2}$ at time $t=0$, where $\psi{1}$ and $\psi{2}$ are particle-in-a-box time-independent energy eigenfunctions [Eq. (2.23)] with $n=1$ and $n=2$, respectively. (a) Find the probability density as a function of time. (b) Show that $|\Psi|^{2}$ oscillates with a period $T=8 m l^{2} / 3 h$. (c) Use a spreadsheet or Mathcad to plot $l|\Psi|^{2}$ versus $x / l$ at each of the times $j T / 8$, where $j=0,1,2, \ldots, 8$.
(a) Since $\hat{H}$ is independent of time, $\Psi$ at any future time will be given by (7.100) with $c{1}=2^{-1 / 2}, c{2}=2^{-1 / 2}$, and all other c's equal to zero. Therefore,

\(
\begin{aligned}
\Psi & =\frac{1}{\sqrt{2}} e^{-i E{1} t / \hbar}\left(\frac{2}{l}\right)^{1 / 2} \sin \frac{\pi x}{l}+\frac{1}{\sqrt{2}} e^{-i E{2} t / \hbar}\left(\frac{2}{l}\right)^{1 / 2} \sin \frac{2 \pi x}{l} \
& =\frac{1}{\sqrt{2}} e^{-i E{1} t / \hbar} \psi{1}+\frac{1}{\sqrt{2}} e^{-i E{2} t / \hbar} \psi{2}
\end{aligned}
\)

We find for the probability density (Prob. 7.47)

\(
\begin{equation}
\Psi^{} \Psi=\frac{1}{2} \psi{1}^{2}+\frac{1}{2} \psi{2}^{2}+\psi{1} \psi{2} \cos \left[\left(E{2}-E{1}\right) t / \hbar\right] \tag{7.102}
\end{}
\)

(b) The time-dependent part of $|\Psi|^{2}$ is the cosine factor in (7.102). The period $T$ is the time it takes for the cosine to increase by $2 \pi$, so $\left(E{2}-E{1}\right) T / \hbar=2 \pi$ and $T=2 \pi \hbar /\left(E{2}-E{1}\right)=8 m l^{2} / 3 h$, since $E{n}=n^{2} h^{2} / 8 m l^{2}$. (c) Using (7.102), the expressions for $\psi{1}$ and $\psi{2}$, and $T=2 \pi \hbar /\left(E{2}-E{1}\right)$, we have
$l|\Psi|^{2}=\sin ^{2}\left(\pi x{r}\right)+\sin ^{2}\left(2 \pi x{r}\right)+2 \sin \left(\pi x{r}\right) \sin \left(2 \pi x{r}\right) \cos (2 \pi t / T)$
where $x{r} \equiv x / l$. With $t=j T / 8$, the graphs are easily plotted for each $j$ value. The plots show that the probability-density maximum oscillates between the left and right sides of the box. Using Mathcad, one can produce a movie of $|\Psi|^{2}$ as time changes (Prob. 7.47). (An online resource that allows one to follow $|\Psi|^{2}$ as a function of time for systems such as the particle in a box or the harmonic oscillator for any chosen initial mixture of stationary states is at www.falstad.com/qm1d; one chooses the mixture by clicking on the small circles at the bottom and dragging on each rotating arrow within a circle.)

Equation (7.100) with the $c{n}$ 's being constant is the general solution of the timedependent Schrödinger equation when $\hat{H}$ is independent of time. For a system acted on by an external time-dependent force, the Hamiltonian contains a time-dependent part: $\hat{H}=\hat{H}^{0}+\hat{H}^{\prime}(t)$, where $\hat{H}^{0}$ is the Hamiltonian of the system in the absence of the external force and $\hat{H}^{\prime}(t)$ is the time-dependent potential energy of interaction of the system with the external force. In this case, we can use the stationary-state time-independent eigenfunctions of $\hat{H}^{0}$ to expand $\Psi$ in an equation like (7.100), except that now the $c{n}$ 's depend on $t$. An example is an atom or molecule exposed to the time-dependent electric field of electromagnetic radiation (light); see Section 9.8.

What determines whether a system is in a stationary state such as (7.99) or a nonstationary state such as (7.100)? The answer is that the history of the system determines its present state. For example, if we take a system that is in a stationary state and expose it to radiation, the time-dependent Schrödinger equation shows that the radiation causes the state to change to a nonstationary state; see Section 9.8.

You might be wondering about the absence from the list of postulates of the Born postulate that $|\Psi(x, t)|^{2} d x$ is the probability of finding the particle between $x$ and $x+d x$. This postulate is a consequence of Postulate 5, as we now show. Equation (3.81) is $\langle B\rangle=\sum{b} P{b} b$, where $P{b}$ is the probability of observing the value $b$ in a measurement of the property $B$ that takes on discrete values. The corresponding equation for the continuous variable $x$ is $\langle x\rangle=\int{-\infty}^{\infty} P(x) x d x$, where $P(x)$ is the probability density for observing various values of $x$. According to Postulate 5, we have $\langle x\rangle=\int{-\infty}^{\infty} \Psi^{*} \hat{x} \Psi d x=\int{-\infty}^{\infty}|\Psi|^{2} x d x$. Comparison of these two expressions for $\langle x\rangle$ shows that $|\Psi|^{2}$ is the probability density $P(x)$.

Chapter 10 gives two further quantum-mechanical postulates that deal with spin and the spin-statistics theorem.