91Ó°ÊÓ

The equation for the time-dependent Schrödinger equation is,

$\mathit{i}\sout{\mathbf{h}}\frac{\mathbf{d}}{\mathbf{d}\mathbf{t}}\mathbf{|}\mathit{Ψ}\left(t\right)\mathbf{>}\mathbf{=}\hat{\mathbf{H}}\mathbf{|}\mathit{Ψ}\left(t\right)\mathbf{>}$

One coordinate, x, is all that is required to describe a bead's location on a ring because the ring is in a two-dimensional plane. The next step is to solve the Schrödinger equation for the wave function in one dimension Ψ(x, t).

The value of x is determined around the circumference.

Here given equation is:

$$\begin{gathered} -\frac{{\sout{h}}^2}{2m}\frac{d^2\mathrm{ψ}}{d{x}^2}=E\mathrm{ψ} \\ \frac{d^2\mathrm{ψ}}{d{x}^2}=-k^2\mathrm{ψ}, \end{gathered}$$

where k =$\frac{\sqrt{2mE}}{\sout{h}}$

$$\begin{gathered} \mathrm{ψ}\left(x\right)=A{e}^{ikx}+B{e}^{-ikx} \\ A{e}^{ikx}{e}^{iKL}+B{e}^{-ikx}{e}^{iKL}=A{e}^{ikx}+B{e}^{-ikx} \\ A{e}^{ikL}+B{e}^{-ikL}=A+B......\left(1\right) \end{gathered}$$

Now,

$$\begin{gathered} A{e}^{i\mathrm{Ï€}/2}{e}^{ikL}+B{e}^{-i\mathrm{Ï€}/2}{e}^{-ikL}=A{e}^{i\mathrm{Ï€}/2}+B{e}^{-i\mathrm{Ï€}/2} \\ iA{e}^{ikL}-iB{e}^{-ikL}=iA-iB \\ A{e}^{ikL}-iB{e}^{-ikL}=A-B.....\left(2\right) \end{gathered}$$

Add equations (1) and (2), and we get,

$2A{e}^{ikL}=2A$

When A= 0 ,$e^{ikL}=1,$$kL=2n\mathrm{Ï€}$, n is an integer.

If A = 0, then $B{e}^{-ikL}=\mathrm{B}$ .

This leads to the same conclusions.

Now, every positive n has two solutions ${\mathrm{ψ}}_n^{+}\left(x\right)=A{e}^{i\left(2n\mathrm{π}x/L\right)}$ and ${\mathrm{ψ}}_n^{-}\left(x\right)=A{e}^{i\left(2n\mathrm{π}x/L\right)}$.

For n = 0 , there is just one solution:

${∫}_0^L{\left|{\mathrm{ψ}}_{Â\pm }\right|}^{2}dx=1A=B=1/\sqrt{L}$

Any other solution will be a linear combination of these as:

$$\begin{gathered} {\mathrm{ψ}}_n^{Â\pm }\left(x\right)=\frac{1}{\sqrt{L}}{e}^{Â\pm i\left(2n\mathrm{Ï€}x/L\right)} \\ {E}_n=\frac{2n^2{\mathrm{Ï€}}^2{\sout{h}}^2}{m{L}^2},n=0,1,2,3,4,... \end{gathered}$$

Hence, we conclude that this theorem fails because:

(1)$\mathrm{ψ}$ does not go to zero at infinity

(2) x is restricted to a finite range

(3) we are unable to determine the constant K