91Ó°ÊÓ

$\text{U(x) =}\left\{\begin{array}{ll}0& \left|x\right|<\frac{\text{1}}{\text{2}}\text{a}\\ ∞& \text{|x| >}\frac{\text{1}}{\text{2}}\text{a}\end{array}\right.$

The wave function on the exterior must be zero, while the wave function on the interior is

$\mathrm{ψ}\left(x\right)=\text{Asin(kx) + Bcos(kx)}$

The wave function must be zero at both edges in this scenario at $\frac{\text{1}}{\text{2}}\text{a}$and $\text{-}\frac{\text{1}}{\text{2}}\text{a}$

$\begin{array}{rcl}& & \mathrm{ψ}\left(\frac{\text{1}}{\text{2}}\text{a}\right)\text{= 0}\\ & & \text{Asin}\left(\text{k}\left(\frac{\text{1}}{\text{2}}\text{a}\right)\right)\text{+ Bcos}\left(\text{k}\left(\frac{\text{1}}{\text{2}}\text{a}\right)\right)\text{=0}\\ & & \text{Asin}\left(\frac{\text{ka}}{\text{2}}\right)\text{+ Bcos}\left(\frac{\text{ka}}{\text{2}}\right)\text{=0..........}\left(1\right)\end{array}$

$\begin{array}{rcl}& & \mathrm{ψ}\left(\frac{\text{1}}{\text{2}}\text{a}\right)\text{= 0}\\ & & \text{Asin}\left(\text{k}\left(\frac{\text{1}}{\text{2}}\text{a}\right)\right)\text{+ Bcos}\left(\text{k}\left(\frac{\text{1}}{\text{2}}\text{a}\right)\right)\text{=0}\\ & & -\text{Asin}\left(\frac{\text{ka}}{\text{2}}\right)\text{+ Bcos}\left(\frac{\text{ka}}{\text{2}}\right)\text{=0..........}\left(2\right)\end{array}$

Add (1) and (2) here,
$$\begin{gathered} \text{2Bcos}\left(\frac{\text{ka}}{\text{2}}\right)\text{= 0} \\ \text{Bcos}\left(\frac{\text{ka}}{\text{2}}\right)\text{= 0} \end{gathered}$$

Where, $\frac{\text{ka}}{\text{2}}\text{=}\frac{{\displaystyle \text{(2n + 1)Ï€}}}{2}\text{;n=0,1,2,3,....}$here n is odd.so,

$$\begin{gathered} \frac{\text{ka}}{\text{2}}\text{=}\frac{n\mathrm{Ï€}}{2} \\ \text{k=}\frac{{\displaystyle n\mathrm{Ï€}}}{{\displaystyle 2}} \end{gathered}$$

Only the cosine function, which is acceptable for odd numbers, is used in the wave function of hence we can substituteaccordingly.

$$\begin{gathered} \mathrm{ψ}\left(x\right)\text{=Bcos}\left(kx\right) \\ \mathrm{ψ}\left(x\right)\text{=Bcos}\left(\frac{n\mathrm{π}x}{{\displaystyle \text{a}}}\right)\text{(n is)odd} \end{gathered}$$

Apply the normalisation condition after that.

$\begin{array}{rcl}& & {∫}_{\text{-}\frac{\text{a}}{\text{2}}}^{\frac{\text{a}}{\text{2}}}{\text{´¥Ïˆ(³æ)´¥}}^{\text{2}}\text{dx = 1}\\ & & {∫}_{\text{-}\frac{\text{a}}{\text{2}}}^{\frac{\text{a}}{\text{2}}}{B}^{\text{2}}{\cos }^{\text{2}}\left(\frac{n\mathrm{Ï€}x}{{\displaystyle \text{a}}}\right)\text{dx = 1}\\ & & B^{\text{2}}\text{×}\frac{\text{a}}{\text{2}}\text{= 1}\\ & & B\text{=}\sqrt{\frac{\text{2}}{\text{a}}}\end{array}$

As a result of this, the wave equation for odd values of n is

$$\begin{gathered} \mathrm{ψ}\left(x\right)=\sqrt{\frac{\text{2}}{\text{a}}\cos }\left(\frac{{\displaystyle n\mathrm{π}x}}{{\displaystyle \text{a}}}\right) \\ \mathrm{ψ}\left(x\right)\text{=Bcos}\left(\frac{n\mathrm{π}x}{{\displaystyle \text{a}}}\right)\text{(n is}\text{)odd} \\ \mathrm{ψ}\left(x\right)=\sqrt{\frac{\text{2}}{\text{a}}\cos }\left(\frac{{\displaystyle n\mathrm{π}x}}{{\displaystyle \text{a}}}\right)\text{(n is}\text{)odd} \end{gathered}$$

So, Energy

$\begin{array}{rcl}& & \text{E =}\frac{{\text{k}}^{\text{2}}{\text{h}}^{\text{2}}}{\text{2m}}\\ & & \text{=}\frac{{\left({\displaystyle \frac{n\mathrm{Ï€}}{a}}\right)}^2{\displaystyle {\text{h}}^2}}{{\displaystyle \text{2m}}}\\ & & \text{=}\frac{{\displaystyle {\text{n}}^{\text{2}}{\text{Ï€}}^{\text{2}}{\text{h}}^{\text{2}}}}{{\displaystyle {\text{2ma}}^{\text{2}}}}\end{array}$

Subtract Equation (1) and (2)

$$\begin{gathered} \text{2Asin}\left(\frac{\text{ka}}{\text{2}}\right)\text{= 0} \\ \text{Asin}\left(\frac{\text{ka}}{\text{2}}\right)\text{= 0} \end{gathered}$$

Equation is valid when $\frac{\text{ka}}{\text{2}}\text{=2²ÔÏ€;²Ô=1,2,3....}$But here n is even

$\text{k =}\frac{n\mathrm{Ï€}}{a}$

The wave function is only in sine, and it is only valid for even numbers n here substitute

$$\begin{gathered} \mathrm{ψ}\left(x\right)=A\sin \left(kx\right) \\ \mathrm{ψ}\left(x\right)=A\sin \left(\frac{{\displaystyle n\mathrm{π}x}}{{\displaystyle \text{a}}}\right)\text{(n is even )} \end{gathered}$$

$\begin{array}{rcl}& & {∫}_{\text{-}\frac{\text{a}}{\text{2}}}^{\frac{\text{a}}{\text{2}}}{\text{´¥Ïˆ(³æ)´¥}}^{\text{2}}\text{dx = 1}\\ & & {∫}_{\text{-}\frac{\text{a}}{\text{2}}}^{\frac{\text{a}}{\text{2}}}{\text{A}}^{\text{2}}{\text{sin}}^{\text{2}}\left(\frac{n\mathrm{Ï€}x}{{\displaystyle \text{a}}}\right)\text{dx = 1}\\ & & {\text{A}}^{\text{2}}\text{×}\frac{\text{a}}{\text{2}}\text{= 1}\\ & & \text{A =}\sqrt{\frac{\text{2}}{\text{a}}}\end{array}$

As a result of this, the wave equation for even values of is

$$\begin{gathered} \mathrm{ψ}\left(x\right)\text{= Asin}\left(\frac{n\mathrm{π}x}{{\displaystyle \text{a}}}\right)\text{(n is even )} \\ \mathrm{ψ}\left(x\right)=\sqrt{\frac{\text{2}}{\text{a}}\text{sin}}\left(\frac{{\displaystyle n\mathrm{π}x}}{{\displaystyle \text{a}}}\right)\text{(n is even)} \end{gathered}$$

For energy,

$\begin{array}{rcl}& & \text{E =}\frac{{\text{k}}^{\text{2}}{\text{h}}^{\text{2}}}{\text{2m}}\\ & & \text{=}\frac{{\left({\displaystyle \frac{n\mathrm{Ï€}}{a}}\right)}^2{\displaystyle {{\displaystyle \text{h}}}^2}}{{\displaystyle \text{2m}}}\\ & & \text{=}\frac{{\displaystyle {\text{n}}^{\text{2}}{\text{Ï€}}^{\text{2}}{\text{h}}^{\text{2}}}}{{\displaystyle {\text{2ma}}^{\text{2}}}}\end{array}$

Therefore the energy of the well is

n is odd$\mathrm{ψ}\left(x\right)=\sqrt{\frac{\text{2}}{\text{a}}\cos }\left(\frac{{\displaystyle n\mathrm{π}x}}{{\displaystyle \text{a}}}\right)$

n is even$\mathrm{ψ}\left(x\right)=\sqrt{\frac{\text{2}}{\text{a}}\text{sin}}\left(\frac{{\displaystyle n\mathrm{π}x}}{{\displaystyle \text{a}}}\right)$

For all n values$\begin{array}{rcl}& & \text{=}\frac{{\displaystyle {\text{n}}^{\text{2}}{\text{Ï€}}^{\text{2}}{\text{h}}^{\text{2}}}}{{\displaystyle {\text{2ma}}^{\text{2}}}}\end{array}$