《Quantum Chemistry》章末总结4

第四章介绍量子力学公设。

Chapter 4 The Postulates and General Principles of Quantum Mechanics

4.1 The State of a System Is Completely Specified by lts Wave Function

Classical mechanics deals with quantities called dynamical variables, such as position, momentum, angular momentum, and energy. A measurable dynamical variable is called an observable.

The classical-mechanical state of a one-body system at any particular time is specified completely by the three position coordinates $(x,y,z)$ and the three momenta or velocities $(v_x,v_y,v_z)$ at that time.

Classical mechanics provides a method for calculating the trajectory of a particle in terms of the forces acting upon the particle through Newton’s equations.

If there are $N$ particles in the system, then it takes $3N$ coordinates and $3N$ velocities to specify the state of the system. We should suspect immediately that this is not going to be so in quantum mechanics because the uncertainty principle tells us that we cannot specify or determine the position and momentum of a particle simultaneously with infinite precision. This leads us to our first postulate of quantum mechanics.

• Postulate 1: The state of a quantum-mechanical system is completely specified by a function $\Psi(\boldsymbol r,t)$ that depends on the coordinates of the particle and on time. This function, called the wave function or state function, has the important property that $\Psi^*(\boldsymbol r,t)\Psi(\boldsymbol r,t)dxdydz$ is the probability that the particle lies in the volume element $dxdydz$ located at $\boldsymbol r$ at time $t$.

Postulate 1 says that the state of a quantum-mechanical system such as two electrons is completely specified by this function and that nothing else is required.

Because the square of the wave function has a probabilistic interpretation, it must satisfy certain physical requirements.

In addition to being normalized, or at least normalizable, because $\Psi^*(\boldsymbol r,t)\Psi(\boldsymbol r,t)d\tau$ is a probability, we require that $\Psi(\boldsymbol r,t)$ and its first spatial derivative be single-valued, continuous, and finite. We summarize these requirements by saying that $\Psi(\boldsymbol r,t)$ must be well behaved.

（ 公设一说明了波函数蕴含了系统的一切性质，波函数的平方表示粒子的概率密度。 ）

4.2 Quantum-Mechanical Operators Represent Classical-Mechanical Variables

• Postulate 2: To every observable in classical mechanics there corresponds a linear operator in quantum mechanics.

（ 公设二是易于理解的，算符作用于波函数将力学量的求解问题转化为本征值问题，每个可观测量都有对应的线性算符。书中说量子力学中都是线性算符，其实不准确，量子力学中也用到一些非线性算符，只不过它们不对应可观测的物理量。

另外，这一条公设将会在4.5小节更严谨地加以修正。 ）

4.3 Observable Quantities Must Be Eigenvalues of Quantum-Mechanical Operators

• Postulate 3: In any measurement of the observable associated with the operator $\hat A$, the only values that will ever be observed are the eigenvalues $a$, which satisfy the eigenvalue equation:

$$
\hat A\Psi_a=a\Psi_a
$$

Generally, an operator will have a set of eigenfunctions and eigenvalues, and we indicate this by writing:

$$
\hat A \Psi_n=a_n\Psi_n
$$

Thus, in any experiment designed to measure the observable corresponding to $\hat A$, the only values we find are $a_1,a_2,a_3,\cdots$, The set of eigenvalues {$a_n$} of an operator $\hat A$ is called the spectrum of $\hat A$.

Because of the equation $\Delta E=h\nu$, we see that the experimentally observed spectrum of the system is intimately related to the mathematical spectrum. This is the motivation for calling the set of eigenvalues of an operator its spectrum, and this is also why the Schrödinger equation is a special eigenvalue equation.

The particle in a box is a bound system, and for bound systems the spectrum is discrete. For an unbound system, the spectrum is continuous.

For other systems, such as a particle in a box with finite walls (Figure 4.1), the spectrum can have both discrete and continuous parts.

（ 公设三与公设二相联系，把力学量求解问题转化为本征值问题。 ）

It is important to realize that the {$\Psi_n$} are very special functions, and it is possible that a system will not be in one of these states. In fact, if we were to measure the energy of each member of a collection of similarly described systems, all in the state described by $\Psi$, then we would observe a distribution of energies, but each member of this distribution will be one of the energies $E_n$ This leads to our fourth postulate.

• Postulate 4: If a system is in a state described by a normalized wave function $\Psi$, then the average value of the observable corresponding to $\hat A$ is given by:

$$
\left\langle a \right\rangle=\int_{-\infin}^{\infin}\Psi^*\hat A\Psi d\tau
$$

Thus, Postulate 3 says that the only value that we measure is the value $a_n$. Often, however, the system is not in a state described by an eigenfunction, and one measures a distribution of values whose average is given by Postulate 4.

（ 如果系统的状态可以用本征函数描述，那么我们通过实验测得的值就是算符对应的力学量；不过有时候系统的状态函数是一个非本征函数，此时公设四指出：实验测得的可观测量为力学量的平均值/期待值。 ）

4.4 The Commutator of Two Operators Plays a Central Role in the Uncertainty Principle

When two operators act sequentially on a function, as in $\hat A\hat Bf(x)$, we apply each operator in turn, working from right to left:

$$
\hat A\hat Bf(x)=\hat A[\hat Bf(x)]=\hat Ah(x)
$$

where $h(x)=\hat Bf(x)$. An important difference between operators and ordinary algebraic quantities is that operators do not necessarily commute:

$$
\hat A\hat Bf(x)=\hat B\hat Af(x) \quad (commutative) \ \hat A\hat Bf(x) \neq \hat B\hat Af(x) \quad (noncommutative)
$$

We called the commutator, $[\hat A,\hat B]$, of $\hat A$ and $\hat B$:

$$
[\hat A,\hat B]=\hat A\hat B-\hat B\hat A
$$

If $[\hat A,\hat B]f(x)=0$ for all $f(x)$ on which the commutator acts, then we write that $[\hat A,\hat B]=0$ and we say that $\hat A$ and $\hat B$ commute.

If two operators do not commute, then their corresponding observable quantities do not have simultaneously well-defined values, and in fact:

$$
\sigma_A^2\sigma_B^2 \ge -\frac 14 \left(\int \psi^* [\hat A,\hat B] \psi dx \right)^2
$$

where $\sigma_A^2$ and $\sigma_B^2$ are:

$$
\sigma_A^2 = \int \psi^(\hat A-\left\langle a \right\rangle)^2 \psi dx \ \sigma_B^2 = \int \psi^(\hat B-\left\langle b \right\rangle)^2 \psi dx
$$

If $\hat A$ and $\hat B$ are the momentum and position operators. In the case:

$$
[\hat A,\hat B]=[\hat P_x,\hat X]=-i\hbar\hat I
$$

where $\hat I$ is the identity operator.

And:

$$
\sigma_p^2\sigma_x^2 \ge -\frac 14(-i\hbar)^2=\frac {\hbar^2}{4}
$$

By taking the square root of both sides, we have:

$$
\sigma_p\sigma_x \ge \frac {\hbar}{2}
$$

which is the Heisenberg uncertainty principle for momentum and position.

4.5 Quantum-Mechanical Operators Must Be Hermitian Operators

We have seen that wave functions, and operators generally, are complex
quantities, but certainly the eigenvalues must be real quantities if they are to correspond to the result of experimental measurement:

$$
\hat A^\psi^=a^\psi^=a\psi^*
$$

where the equality $a^*=a$ recognizes that $a$ is real. So:

$$
\int \psi\hat A^*\psi^*dx = a\int\psi\psi^dx = a =\int \psi^ \hat A\psi dx
$$

The operator $\hat A$ must satisfy the equation to assure that its eigenvalues are real. An operator that satisfies the equation for any well-behaved function is called a Hermitian operator. Thus, we can write the definition of a Hermitian operator as an operator that satisfies the relation:

$$
\int_{-\infin}^{\infin} f^\hat Afdx = \int_{-\infin}^{\infin} f {\hat A}^ f^* dx
$$

Hermitian operators have real eigenvalues. Postulate 2 should be modified to read as follows:

• Postulate 2’: To every observable in classical mechanics there corresponds a linem; Hermitian operator in quantum mechanics.

A more general definition of a Hermitian operator is given by:

$$
\int_{-\infin}^{\infin} dx f_m^(x) \hat A f_n(x) = \int_{-\infin}^{\infin}dx f_n(x) {\hat A}^ f^*_m(x)
$$

where $f_n(x)$ and $f_m(x)$ are any two well-behaved functions. We shall use this definition quite often.

（ 厄米算符对应的本征值为实数，所以很容易理解为什么其在量子力学中十分重要了，因为本征值为实数才有物理意义。 ）

This is a good time to introduce a notation that is extremely widely used. Let {$f_n(x)$} be some set of well-behaved functions that we label by an integer $n$. These functions might, but not necessarily, be the eigenfunctions of some operator $\hat O$.

In this new notational scheme, we shall express $f_n(x)$ as $|n\rangle$. Instead of denoting the complex conjugate of $f_n(x)$ by $|n\rangle^$, we denote it by $\langle n|$. The integral of $f_n^(x)f_n(x)$ in this notation is expressed as $\langle n|n \rangle$ , or:

$$
\int_{-\infin}^{\infin} dx f_n^*(x)f_n(x)= \langle n|n \rangle
$$

Normalization corresponds to writing $\langle n|n \rangle=1$. More generally:

$$
\int_{-\infin}^{\infin} dx f_m^*(x)f_n(x)= \langle m|n \rangle
$$

Continuing, for an operator $\hat A$ we write:

$$
\int_{-\infin}^{\infin} dx f_m^*(x) \hat A f_n(x)= \langle m|\hat A |n \rangle
$$

and so $\hat A$ is a Hermitian operator if:

$$
\langle m|\hat A |n \rangle = \langle n|\hat A |m \rangle^*
$$

This notation is due to the British physicist Paul Dirac. The quantities $|n\rangle$ are called kets, and the $\langle m|$ are called bras. This nomenclature arises from the fact that integrals are denoted by $\underline{bra} \ c\ \underline{kets}$, $\langle m|\hat A|n \rangle$. The notational scheme is called Dirac notation or bracket notation.

4.6 The Eigenfunctions of Hermitian Operators Are Orthogonal

We have been led naturally to the definition and use of Hemiitian operators by requiring that quantum-mechanical operators have real eigenvalues. Not only are the eigenvalues of Hermitian operators real, but their eigenfunctions satisfy a rather special condition as well. Consider the two eigenvalue equations:

$$
\hat A\psi_n = a_n\psi_n \quad \hat A\psi_m = a_m\psi_m
$$

We multiply the first of equations by $\psi_m^*$ and integrate; then we take the complex conjugate of the second, multiply by $\psi_n$ and integrate to obtain:

$$
\langle m|\hat A|n \rangle = a_n\langle m|n \rangle \ \langle n|\hat A|m \rangle^* = a_m^*\langle m|n \rangle
$$

and:

$$
\langle m|\hat A|n \rangle - \langle n|\hat A|m \rangle^=(a_n-a_m^)\langle m|n \rangle
$$

Because $\hat A$ is Hermitian, so we have:

$$
(a_n-a_m^*)\langle m|n \rangle =0
$$

When $n=m$, the integral is unity by normalization and so we have:

$$
a_n=a_n^*
$$

which is just another proof that the eigenvalues are real.

When $n \neq m$ and if the system is nondegenerate, we have:

$$
\langle m|n \rangle =0
$$

A set of eigenfunctions that satisfies the condition in the equation is said to be orthogonal. We have just proved that the eigenfunctions of a Hermitian operator are orthogonal, at least for a nondegenerate system.

A set of functions that are both normalized and orthogonal to each other is called an orthonormal set. We can express the condition of orthonormality by writing:

$$
\langle m|n \rangle = \delta_{nm}
$$

where:

$$
\delta_{nm}=\left{\begin{aligned} 1 \quad m=n \ 0 \quad m\neq n \end{aligned} \right .
$$

The symbol $\delta_{nm}$ occurs frequently and is called the Kronecker delta.

Let’s consider the case in which two states, described by $\psi_1$ and $\psi_2$, have the same eigenvalue $a_1$:

$$
\hat A|\psi_1\rangle=a_1|\psi_1\rangle \quad \hat A|\psi_2\rangle=a_1|\psi_2\rangle
$$

Now let’s consider a linear combination of $\psi_1$ and $\psi_2$, say $\phi =c_1\psi_1+c_2\psi_2$. Then:

$$
\hat A|\phi\rangle =c_1 \hat A|\psi_1\rangle+c_2\hat A|\psi_2\rangle = a_1(c_1|\psi_1\rangle +c_2|\psi_2\rangle)=a_1|\phi\rangle
$$

Thus, we see that if $\psi_1$ and $\psi_2$ describe a two-fold degenerate state with eigenvalue $a_1$, then any linear combination of $\psi_1$ and $\psi_2$ is also an eigenfunction with eigenvalue $a_1$.

It is convenient to choose two linear combinations of $\psi_1$ and $\psi_2$, call them $\phi_1$ and $\phi_2$, such that:

$$
\langle \phi_1|\phi_2 \rangle =0
$$

（ 对于一个简并系统，$\psi_1$和$\psi_2$的线性组合仍然具有相同的本征值，所以我们可以通过一些手段让线性组合后的结果满足正交归一，这种方法就是线性代数里讲的Gram-Schmidt正交化。 ）

So even if there is a degeneracy, we can construct the eigenfunctions of a Hermitian operator such that they are orthonormal and say that they form an orthonormal set.

4.7 If Two Operators Commute, They Have a Mutual Set of Eigenfunctions

Suppose that two operators $\hat A$ and $\hat B$ have the same set of eigenfunctions, so that we have:

$$
\hat A|\phi_n\rangle = a_n|\phi_n\rangle \quad \hat B|\phi_n\rangle = b_n|\phi_n\rangle
$$

We shall prove that if two operators have the same set of eigenfunctions, then they necessarily commute. To prove this, we must show that:

$$
[\hat A,\hat B]f(x)=0
$$

for an arbitrary function $f(x)$. We can expand $f(x)$ in terms of the complete set of eigenfunctions of $\hat A$ and $\hat B$, and write:

$$
f(x)=\sum_nc_n\phi_n(x)
$$

Substitute this expansion into the last euation, then we obtain:

$$
[\hat A,\hat B]f(x)=\sum_nc_n(a_nb_n-b_na_n)\phi_n(x)=0
$$

Because $f(x)$ is arbitrary, $[\hat A,\hat B]=0$. Thus, we see that $\hat A$ and $\hat B$ commute if they have the same set of eigenfunctions.

The converse is also true; if $\hat A$ and $\hat B$ commute, then they have a mutual set of eigenfunctions.

4.8 The Probability of Obtaining a Certain Value of an Observable in a Measurement Is Given by a Fourier Coefficient

Consider a fairly arbitrary function $f(x)$. We assume that if {$\psi_n(x)$} is some orthonormal set defined over the same interval as $f(x)$ and satisfying the same boundary conditions as $f(x)$, then it is possible to write $f(x)$ as:

$$
f(x)=\sum_{n=1}^{\infin}c_n\psi_n(x)
$$

A set of functions such as the $\psi_n(x)$ here is said to be complete if the equation holds for a suitably arbitrary function $f(x)$. It is generally difficult to prove completeness, and in practice one usually assumes that the orthonormal set associated with some Hermitian operator is complete. Then we find:

$$
\langle m|f(x) \rangle =c_m
$$

All the terms in the summation equal zero except for the one term where $n=m$.

The expansion of a function in terms of an orthonormal set is an important and useful technique in many branches of physics and chemistry. The expansion of a function in terms of an orthonormal set is called a Fourier expansion or a Fourier series. The coefficients $c_n$ in the expansion are called Fourier coefficients.

Postulate 4 tells us how to calculate the average in a series of measurements:

$$
\langle a \rangle = \langle \psi(x)|\hat A|\psi(x) \rangle
$$

What is the probability of obtaining the particular result $a_n$ in a single measurement? Well, we find that we can interpret $|c_n|^2$ as the probability of observing $E_n$ when carrying out a measurement on the system, or:

$$
probability\ of\ observing\ E_n = |c_n|^2
$$

4.9 The Time Dependence of Wave Functions Is Governed by the Time-Dependent Schrödinger Equation

• Postulate 5: The wave function or state function of a system evolves in time according to the time-dependent Schrödinger equation:

$$
\hat H \Psi(x,t)=i\hbar \frac {\partial \Psi}{\partial t}
$$

For most systems that we shall study in this book, $\hat H$ does not contain time explicitly, and in this case we can apply the method of separation of variables and write:

$$
\Psi(x,t)=\psi(x)f(t)
$$

$$
\Psi(x,t)=\psi(x)e^{-i\omega t}
$$

It is interesting to note that the equation oscillates harmonically in time and is characteristic of wave motion. Yet the time-dependent Schrödinger equation does not have the same form as a classical wave equation. Nevertheless, the Schrödinger equation does have wavelike solutions, which is one reason why quantum mechanics is sometimes called wave mechanics.

We can use the time-dependent Schrödinger equation to derive an explicit expression for the time dependence of the average value of an operator. Start with:

$$
\langle A\rangle = \int d\tau \Psi^*(x,t)\hat A(x,t)\Psi(x,t)
$$

Differentiate with respect to $t$ to obtain:

$$
\frac {d\langle A \rangle}{dt} = \int d\tau \left(\frac {i\hat H\Psi^}{\hbar} \right)\hat A\Psi + \langle {\frac {\partial \hat A}{\partial t}} \rangle + \int d\tau \Psi^\hat A \left(\frac {i\hat H\Psi}{\hbar} \right)
$$

Rewrite the equation:

$$
\begin{aligned} \frac {d\langle A \rangle}{dt} &=\frac {i}{\hbar}\int d\tau \Psi^\hat H\hat A\Psi - \frac {i}{\hbar}\int d\tau \Psi^\hat A\hat H\Psi + \left\langle {\frac {\partial \hat A}{\partial t}} \right\rangle \[3mm] &=\frac {i}{\hbar} \langle \Psi|[\hat H,\hat A]|\Psi \rangle+ \left\langle {\frac {\partial \hat A}{\partial t}} \right\rangle \end{aligned}
$$

In fact, if we just undo the indicated integrations in the equation, we have the quantum-mechanical equation of motion of the operator $\hat A$ itself:

$$
\frac {d\hat A}{dt} = \frac {i}{\hbar}[\hat H,\hat A]+\frac {\partial \hat A}{\partial t}
$$

Note that if $\hat A$ commutes with $\hat H$ if and does not depend explicitly on
time, then $d\langle \hat A \rangle/dt=0$, which means that $\langle \hat A\rangle$ is a constant of motion, or is conserved.

4.10 Quantum Mechanics Can Describe the Two-Slit Experiment

You should be convinced by now that if a system is some superposition state or mixed state described by:

$$
\Psi(x,t)=\sum_nc_n\psi_n(x)e^{-iE_nt/\hbar}
$$

then a measurement of the energy will yield one of the values $E_n$ with a probability $|c_n|^2$.

Suppose we measure the energy at some time $t_0$ and obtain the value $E_3$. Now suppose that we are able to measure the energy immediately after $t_0$, say at $t_0+\epsilon$, where $\epsilon$ is vanishingly small. What value of the energy will we observe? Well, we just found it to be $E_3$, and unless we are willing to allow the system to change its state essentially infinitely rapidly (which we are not), then we must observe the value $E_3$ again.

We say that the wave function has “collapsed” from the superposition to the single state. For classical systems, a measurement of some property of the system does not alter the system in any significant way, but for quantum-mechanical systems, the measurement process has a profound effect.

（ 在我们观测系统之前，系统处于叠加态，但是我们一旦观测系统，系统就会坍缩为一个确定的状态。 ）

The interpretation of the measurement process that we have given here is due primarily to Bohr and Heisenberg, and is called the Copenhagen interpretation. The one entitled In Search of Schrödinger’s Cat is based upon a thought experiment proposed by Schrödinger in which a cat is in a state that is a superposition of a live cat and a dead cat, and just what such a state actually means.

（ 哥本哈根诠释实际上“没有做出诠释”，因为它只是在描述实验现象，目前也无法给“观测”下一个严谨的数学上的定义。此时，很难不让人想起Feynman的那句经典名言:“On the other hand, I think I can safely say that nobody understands quantum mechanics.” ）

At last, we summarizes our set of postulates:

总结

本章介绍了量子力学的基本公设，注意不同内容层次、不同内容深度的教材陈列的基本公设可能也会有差别。例如本书中的五大公设分别是：1.波函数公设和波恩概率诠释、2.力学量使用线性厄米算符表示、3.力学量的测量值为算符的本征值、4.力学量的期待值计算、5.波函数满足薛定谔方程随时间演化。不过，在不少其他教材中，并没有本书所说的公设4，而是全同粒子公设；还有的书中把算符对易作为基本公设之一。

另外，我省略了不少例题，实际上量子力学的学习是离不开习题训练的，多少得做一做。