Up to now we have restricted ourselves to one-dimensional, one-particle systems. The operator formalism developed in the last section allows us to extend our work to threedimensional, many-particle systems. The time-dependent Schrödinger equation for the time development of the state function is postulated to have the form of Eq. (1.13):
\(
\begin{equation}
i \hbar \frac{\partial \Psi}{\partial t}=\hat{H} \Psi \tag{3.42}
\end{equation}
\)
The time-independent Schrödinger equation for the energy eigenfunctions and eigenvalues is
\(
\begin{equation}
\hat{H} \psi=E \psi \tag{3.43}
\end{equation}
\)
which is obtained from (3.42) by taking the potential energy as independent of time and applying the separation-of-variables procedure used to obtain (1.19) from (1.13).
For a one-particle, three-dimensional system, the classical-mechanical Hamiltonian is
\(
\begin{equation}
H=T+V=\frac{1}{2 m}\left(p{x}^{2}+p{y}^{2}+p_{z}^{2}\right)+V(x, y, z) \tag{3.44}
\end{equation}
\)
Introducing the quantum-mechanical operators [Eq. (3.24)], we have for the Hamiltonian operator
\(
\begin{equation}
\hat{H}=-\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}\right)+V(x, y, z) \tag{3.45}
\end{equation}
\)
The operator in parentheses in (3.45) is called the Laplacian operator $\nabla^{2}$ (read as "del squared"):
\(
\begin{equation}
\nabla^{2} \equiv \frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}} \tag{3.46}
\end{equation}
\)
The one-particle, three-dimensional, time-independent Schrödinger equation is then
\(
\begin{equation}
-\frac{\hbar^{2}}{2 m} \nabla^{2} \psi+V \psi=E \psi \tag{3.47}
\end{equation}
\)
Now consider a three-dimensional system with $n$ particles. Let particle $i$ have mass $m{i}$ and coordinates $\left(x{i}, y{i}, z{i}\right)$, where $i=1,2,3, \ldots, n$. The kinetic energy is the sum of the kinetic energies of the individual particles:
\(
T=\frac{1}{2 m{1}}\left(p{x{1}}^{2}+p{y{1}}^{2}+p{z{1}}^{2}\right)+\frac{1}{2 m{2}}\left(p{x{2}}^{2}+p{y{2}}^{2}+p{z{2}}^{2}\right)+\cdots+\frac{1}{2 m{n}}\left(p{x{n}}^{2}+p{y{n}}^{2}+p{z_{n}}^{2}\right)
\)
where $p{x{i}}$ is the $x$ component of the linear momentum of particle $i$, and so on. The kineticenergy operator is
\(
\begin{gather}
\hat{T}=-\frac{\hbar^{2}}{2 m{1}}\left(\frac{\partial^{2}}{\partial x{1}^{2}}+\frac{\partial^{2}}{\partial y{1}^{2}}+\frac{\partial^{2}}{\partial z{1}^{2}}\right)-\cdots-\frac{\hbar^{2}}{2 m{n}}\left(\frac{\partial^{2}}{\partial x{n}^{2}}+\frac{\partial^{2}}{\partial y{n}^{2}}+\frac{\partial^{2}}{\partial z{n}^{2}}\right) \
\hat{T}=-\sum{i=1}^{n} \frac{\hbar^{2}}{2 m{i}} \nabla{i}^{2} \tag{3.48}\
\nabla{i}^{2} \equiv \frac{\partial^{2}}{\partial x{i}^{2}}+\frac{\partial^{2}}{\partial y{i}^{2}}+\frac{\partial^{2}}{\partial z_{i}^{2}} \tag{3.49}
\end{gather}
\)
We shall usually restrict ourselves to cases where the potential energy depends only on the $3 n$ coordinates:
\(
V=V\left(x{1}, y{1}, z{1}, \ldots, x{n}, y{n}, z{n}\right)
\)
The Hamiltonian operator for an $n$-particle, three-dimensional system is then
\(
\begin{equation}
\hat{H}=-\sum{i=1}^{n} \frac{\hbar^{2}}{2 m{i}} \nabla{i}^{2}+V\left(x{1}, \ldots, z_{n}\right) \tag{3.50}
\end{equation}
\)
and the time-independent Schrödinger equation is
\(
\begin{equation}
\left[-\sum{i=1}^{n} \frac{\hbar^{2}}{2 m{i}} \nabla{i}^{2}+V\left(x{1}, \ldots, z_{n}\right)\right] \psi=E \psi \tag{3.51}
\end{equation}
\)
where the time-independent wave function is a function of the $3 n$ coordinates of the $n$ particles:
\(
\begin{equation}
\psi=\psi\left(x{1}, y{1}, z{1}, \ldots, x{n}, y{n}, z{n}\right) \tag{3.52}
\end{equation}
\)
The Schrödinger equation (3.51) is a linear partial differential equation.
As an example, consider a system of two particles interacting so that the potential energy is inversely proportional to the distance between them, with $c$ being the proportionality constant. The Schrödinger equation (3.51) becomes
FIGURE 3.1 An infinitesimal box-shaped region located at $x^{\prime}, y^{\prime}, z^{\prime}$.
\(
\begin{gather}
{\left[-\frac{\hbar^{2}}{2 m{1}}\left(\frac{\partial^{2}}{\partial x{1}^{2}}+\frac{\partial^{2}}{\partial y{1}^{2}}+\frac{\partial^{2}}{\partial z{1}^{2}}\right)-\frac{\hbar^{2}}{2 m{2}}\left(\frac{\partial^{2}}{\partial x{2}^{2}}+\frac{\partial^{2}}{\partial y{2}^{2}}+\frac{\partial^{2}}{\partial z{2}^{2}}\right)\right.} \
\left.+\frac{c}{\left[\left(x{1}-x{2}\right)^{2}+\left(y{1}-y{2}\right)^{2}+\left(z{1}-z{2}\right)^{2}\right]^{1 / 2}}\right] \psi=E \psi \tag{3.53}\
\psi=\psi\left(x{1}, y{1}, z{1}, x{2}, y{2}, z{2}\right)
\end{gather}
\)
Although (3.53) looks formidable, we shall solve it in Chapter 6.
For a one-particle, one-dimensional system, the Born postulate [Eq. (1.15)] states that $\left|\Psi\left(x^{\prime}, t\right)\right|^{2} d x$ is the probability of observing the particle between $x^{\prime}$ and $x^{\prime}+d x$ at time $t$, where $x^{\prime}$ is a particular value of $x$. We extend this postulate as follows. For $a$ three-dimensional, one-particle system, the quantity
\(
\begin{equation}
\left|\Psi\left(x^{\prime}, y^{\prime}, z^{\prime}, t\right)\right|^{2} d x d y d z \tag{3.54}
\end{equation}
\)
is the probability of finding the particle in the infinitesimal region of space with its $x$ coordinate lying between $x^{\prime}$ and $x^{\prime}+d x$, its $y$ coordinate lying between $y^{\prime}$ and $y^{\prime}+d y$, and its $z$ coordinate between $z^{\prime}$ and $z^{\prime}+d z$ (Fig. 3.1). Since the total probability of finding the particle is 1 , the normalization condition is
\(
\begin{equation}
\int{-\infty}^{\infty} \int{-\infty}^{\infty} \int_{-\infty}^{\infty}|\Psi(x, y, z, t)|^{2} d x d y d z=1 \tag{3.55}
\end{equation}
\)
For a three-dimensional, n-particle system, we postulate that
\(
\begin{equation}
\left|\Psi\left(x{1}^{\prime}, y{1}^{\prime}, z{1}^{\prime}, x{2}^{\prime}, y{2}^{\prime}, z{2}^{\prime}, \ldots, x{n}^{\prime}, y{n}^{\prime}, z{n}^{\prime}, t\right)\right|^{2} d x{1} d y{1} d z{1} d x{2} d y{2} d z{2} \ldots d x{n} d y{n} d z{n} \tag{3.56}
\end{equation}
\)
is the probability at time $t$ of simultaneously finding particle 1 in the infinitesimal rectangular box-shaped region at $\left(x{1}^{\prime}, y{1}^{\prime}, z{1}^{\prime}\right)$ with edges $d x{1}, d y{1}, d z{1}$, particle 2 in the infinitesimal box-shaped region at $\left(x{2}^{\prime}, y{2}^{\prime}, z{2}^{\prime}\right)$ with edges $d x{2}, d y{2}, d z{2}, \ldots$, and particle $n$ in the infinitesimal box-shaped region at $\left(x{n}^{\prime}, y{n}^{\prime}, z{n}^{\prime}\right)$ with edges $d x{n}, d y{n}, d z{n}$. The total probability of finding all the particles is 1 , and the normalization condition is
\(
\begin{equation}
\int{-\infty}^{\infty} \int{-\infty}^{\infty} \int{-\infty}^{\infty} \cdots \int{-\infty}^{\infty} \int{-\infty}^{\infty} \int{-\infty}^{\infty}|\Psi|^{2} d x{1} d y{1} d z{1} \cdots d x{n} d y{n} d z{n}=1 \tag{3.57}
\end{equation}
\)
It is customary in quantum mechanics to denote integration over the full range of all the coordinates of a system by $\int d \tau$. A shorthand way of writing (3.55) or (3.57) is
\(
\begin{equation}
\int|\Psi|^{2} d \tau=1 \tag{3.58}
\end{equation}
\)
Although (3.58) may look like an indefinite integral, it is understood to be a definite integral. The integration variables and their ranges are understood from the context.
For a stationary state, $|\Psi|^{2}=|\psi|^{2}$, and
\(
\begin{equation}
\int|\psi|^{2} d \tau=1 \tag{3.59}
\end{equation}
\)