Particle in a box

A particle in a box is a mathematical model used to illustrate the key principles of quantum mechanics in a simplified system involving the particle moving freely within the box.

1-D box

Consider the case of a free particle of mass $m$ moving between $x=0$ and $x=a$ , where $a> 0$ , along the $x$ -axis of a one-dimensional box. Since a free particle is under the influence of no potential energy, we set $V(x)=0$ in eq44 to give

$-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}\psi(x)=E\psi(x)\;\;\;\;\;\;\;\;45a$

where the boundary conditions impose the restriction $\psi(0)=\psi(a)=0$ such that the probability of finding the particle $\vert\psi(x)\vert^2$ outside the box is zero.

The general non-trivial solution of eq45a is $\psi(x)=Asinkx+bcoskx$ , where $A$ and $B$ are non-zero constants. Applying the first boundary condition $\psi(0)=0$ , $B$ must be zero, which results in the specific solution $\psi(x)=Asinkx$ . Imposing the second boundary condition $\psi(a)=0$ , we have $\psi(a)Asinka=0$ . This implies that $ka=n\pi$ , where $n=1,2,\cdots$

Question

Can $n$ be zero or negative?

Answer

If $n=0$ , then $k=0$ , and $\psi(x)=Asin(0)=0$ for all $x$ . This would mean that the probability of finding the particle anywhere within the box is zero, which is not a physically meaningful solution. Negative values of $n$ would give the same probability density as positive values of $n$ . Therefore, including negative integers for $n$ would not yield any new, physically distinct solutions. The set of positive integers $n=1,2,\cdots$ is sufficient to describe all possible states of the particle in the box.

It follows that the wavefunction for a particle in a 1-D box is

$\psi(x)=Asin\frac{n\pi x}{a}\;\;\;\;\;\;\;\;45b$

Solving eq45a using eq45b yields:

$E_n=\frac{n^2h^2}{8ma^2}\;\;\;\;n=1,2,\cdots\;\;\;\;45c$

Eq45c shows that the energy of the particle is quantised, with $n$ being the quantum number.

Question

Normalise the wavefunction in eq45b.

Answer

$\int_0^a\psi^*(x)\psi(x)dx=A^2\int_0^asin^2\frac{n\pi x}{a}dx=1$

Setting $u=\frac{n\pi x}{a}$ , with $du=\frac{n\pi}{a}dx$ , and noting that $sin^2u=\frac{1-cos2u}{2}$ , gives

$A^2\frac{a}{n\pi}\int_0^{n\pi}\frac{1-cos2u}{2}du=1$

Therefore, $A=\sqrt{\frac{2}{a}}$ and $\psi(x)=\sqrt{\frac{2}{a}}sin\frac{n\pi x}{a}$ .

3-D box

For a three-dimensional box of lengths $a$ , $b$ and $c$ , the Hamiltonian is given by eq45. Similarly, we set $V(x,y,z)=0$ , resulting in:

$-\frac{\hbar^{2}}{2m}\biggr$\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}\biggr$\psi(x)=E\psi(x)\;\;\;\;\;\;\;\;45d$

Since the motion of the particle in the box along each axis is independent, we can approximate the wavefunction as a product of three functions, each depending on one of the independent coordinates $x$ , $y$ and $z$ :

$\Psi(x,y,z)=\psi_1(x)\psi_2(y)\psi_3(z)\;\;\;\;\;\;\;\;45e$

Using the separation of variables method, we substitute eq45e into eq45d and divide through by eq45e to yield:

$-\frac{\hbar^{2}}{2m\psi_1(x)}\frac{\partial^{2}\psi_1(x)}{\partial x^{2}}-\frac{\hbar^{2}}{2m\psi_2(y)}\frac{\partial^{2}\psi_2(y)}{\partial y^{2}}-\frac{\hbar^{2}}{2m\psi_3(z)}\frac{\partial^{2}\psi_3(z)}{\partial z^{2}}\psi(x)=E\;\;\;\;\;\;\;\;45f$

Since $x$ , $y$ and $z$ are independent variables, the value of each term on RHS of eq45f varies independently, each producing a constant corresponding to the energy in that dimension, with the sum equaling $E$ , i.e. $E_x+E_y+E_z=E$ . In other words, eq45f can be solved by evaluating the following three equations separately:

$-\frac{\hbar^{2}}{2m\psi_1(x)}\frac{\partial^{2}\psi_1(x)}{\partial x^{2}}=E_x$

$-\frac{\hbar^{2}}{2m\psi_2(y)}\frac{\partial^{2}\psi_2(y)}{\partial x^{2}}=E_y$

$-\frac{\hbar^{2}}{2m\psi_3(z)}\frac{\partial^{2}\psi_3(z)}{\partial x^{2}}=E_z$

The respective boundary conditions are

$\psi_1(0)=\psi_1(a)=0\;\;\;\forall y,z$

$\psi_2(0)=\psi_2(b)=0\;\;\;\forall x,z$

$\psi_3(0)=\psi_3(c)=0\;\;\;\forall x,y$

Therefore, the expressions for the three wavefunctions can be derived using the same logic as for the 1-D case, giving:

$\psi_1(x)=\sqrt{\frac{2}{a}}sin\frac{n_x\pi x}{a}\;\;\;\;n_x=1,2,\cdots$

$\psi_2(y)=\sqrt{\frac{2}{b}}sin\frac{n_y\pi y}{b}\;\;\;\;n_y=1,2,\cdots$

$\psi_3(z)=\sqrt{\frac{2}{c}}sin\frac{n_z\pi z}{c}\;\;\;\;n_z=1,2,\cdots$

with

$\Psi(x,y,z)=\sqrt{\frac{8}{abc}}sin\frac{n_x\pi x}{a}sin\frac{n_y\pi y}{b}sin\frac{n_z\pi z}{c}\;\;\;\;\;\;\;\;45g$

and

$E_{n_xn_yn_z}=\frac{h^2}{8m}\biggr$\frac{n_x^2}{a^2}+\frac{n_y^2}{b^2}+\frac{n_z^2}{c^2}\biggr$\;\;\;\;n_x,n_y,n_z=1,2,\cdots\;\;\;\;45h$

Since the energy of a particle moving in a 1-D (or 3-D) box is purely kinetic, eq45c (or eq45h) describes the translational energy of that particle in the box. In general, the wavefunction and energy equations of a particle in 1-D and 3-D boxes have applications in testing the validity of the position and momentum operators, as well as in perturbation theory and statistical thermodynamics.

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