Question

The one-dimensional time-independent Schrödinger equation is
.
a/ A particle of mass m is contained in a one-dimensional box of width a. The potential energy U(x) is infinite at the walls of the box (x = 0 and x = a) and zero in between (0 < x < a).
Show that the solutions have the form: U(x)=Csin(n.pi.x/a) . Find the constant C.
b/ For the case n = 3, find the probability that the particle will be located in the region a/3 <x< 2a/3
.
c/ Sketch the wave-functions and the corresponding probability density distributions for the cases n = 1, 2 and 3.

Answer 1

Answer on question 60941

The one-dimensional time-independent Schrödinger equation is

a/ A particle of mass $m$ is contained in a one-dimensional box of width $a$. The potential energy $U(x)$ is infinite at the walls of the box ($x = 0$ and $x = a$) and zero in between ($0 < x < a$).

Show that the solutions have the form: $U(x) = C\sin(n \cdot pi \cdot x / a)$. Find the constant $C$.

b/ For the case $n = 3$, find the probability that the particle will be located in the region $a/3 < x < 2a/3$.

c/ Sketch the wave-functions and the corresponding probability density distributions for the cases $n = 1, 2$ and 3.

Solution

$$\left(\frac{-h^2}{2m} \frac{d^2}{dx^2} + U(x)\right)\varphi(x) = E\varphi(x) \quad (*)$$

a) Let check that the solutions have the form:

$$\varphi_n(x) = C_n \sin\left(\frac{n\pi x}{a}\right)$$

The particle can be located only between two walls (due to infinity of potential outside). So we have additional condition $\varphi_n(0) = \varphi_n(a) = 0$.

Using (*) we write:

$$\frac{-h^2}{2m} \frac{d^2}{dx^2} C_n \sin\left(\frac{n\pi x}{a}\right) = \frac{h^2}{2m} \left(\frac{\pi n}{a}\right)^2 \varphi_n(x) = E\varphi(x)$$

As we can from the last, the solutions have the needed form.

For finding $C_n$ we use the condition:

$$\begin{array}{l} \int_{-\infty}^{\infty} |\varphi_n(x)|^2 dx = 1 \rightarrow C_n^2 \int_{-a}^{a} \left| \sin\left(\frac{n\pi x}{a}\right) \right|^2 dx = \frac{C_n^2}{2} \int_{-a}^{a} \left(1 - \cos\left(\frac{2n\pi x}{a}\right)\right) dx = a C_n^2 \\ \rightarrow C_n = \frac{1}{\sqrt{a}} \end{array}$$

b) $P\left(x \in \left[\frac{a}{3}; \frac{2a}{3}\right]\right) = \int_{\frac{a}{3}}^{2\frac{a}{3}} |\varphi_3(x)|^2 dx = \frac{1}{a} \int_{\frac{a}{3}}^{2\frac{a}{3}} \left| \sin\left(\frac{3\pi x}{a}\right) \right|^2 dx = \frac{1}{2a} \int_{\frac{a}{3}}^{2\frac{a}{3}} \left(1 - \cos\left(\frac{6\pi x}{a}\right)\right) dx = \frac{1}{6}$

c) (for sketching we take $a = 1$)

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