Question

A particle is bound by a potential energy of the form

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

This differs from the infinite well of Section 5.5in being symmetric about, which implies that the probability densities must also he symmetric. Noting that either sine or cosine would fit this requirement and could be made continuous with the zero wave function outside the well. Determine the allowed energies and corresponding normalised wave functions for this potential well.

Short answer

The allowed energies and corresponding normalised wave functions for this well.

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

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

n For all valuesrole="math" localid="1660109339767" $\begin{array}{rcl}& & \text{E=}\frac{{\displaystyle {\text{n}}^{\text{2}}{\text{π}}^{\text{2}}{\text{h}}^{\text{2}}}}{{\displaystyle {\text{2ma}}^{\text{2}}}}\end{array}$

Step-by-step solution

Given information.

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

Step 2: Wave function on the interior.

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

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

Energy.

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

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

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

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

$$\begin{gathered} \frac{\text{ka}}{\text{2}}\text{=}\frac{n\pi }{2} \\ \text{k=}\frac{{\displaystyle n\pi }}{{\displaystyle 2}} \end{gathered}$$

Step 4: is odd.

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

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

Energy when is odd.

Apply the normalisation condition after that.

$\begin{array}{rcl}& & {\int }_{\text{-}\frac{\text{a}}{\text{2}}}^{\frac{\text{a}}{\text{2}}}{\text{|ψ(x)|}}^{\text{2}}\text{dx = 1}\\ & & {\int }_{\text{-}\frac{\text{a}}{\text{2}}}^{\frac{\text{a}}{\text{2}}}{B}^{\text{2}}{\cos }^{\text{2}}\left(\frac{n\pi x}{{\displaystyle \text{a}}}\right)\text{dx = 1}\\ & & B^{\text{2}}\text{×}\frac{\text{a}}{\text{2}}\text{= 1}\\ & & B\text{=}\sqrt{\frac{\text{2}}{\text{a}}}\end{array}$

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

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

So, Energy

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

Subtract Equation (1) and (2)

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 $\frac{\text{ka}}{\text{2}}\text{=2nπ;n=1,2,3....}$But here n is even

$\text{k =}\frac{n\pi }{a}$

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

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

Energy when is even.

$\begin{array}{rcl}& & {\int }_{\text{-}\frac{\text{a}}{\text{2}}}^{\frac{\text{a}}{\text{2}}}{\text{|ψ(x)|}}^{\text{2}}\text{dx = 1}\\ & & {\int }_{\text{-}\frac{\text{a}}{\text{2}}}^{\frac{\text{a}}{\text{2}}}{\text{A}}^{\text{2}}{\text{sin}}^{\text{2}}\left(\frac{n\pi x}{{\displaystyle \text{a}}}\right)\text{dx = 1}\\ & & {\text{A}}^{\text{2}}\text{×}\frac{\text{a}}{\text{2}}\text{= 1}\\ & & \text{A =}\sqrt{\frac{\text{2}}{\text{a}}}\end{array}$

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

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

For energy,

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

Therefore the energy of the well is

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

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

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