MatinSAR
- 673
- 204
- Homework Statement
- Consider a particle of mass $m$ confined in a one-dimensional infinite square well potential centered at ##x=0##, defined as:$$ V(x) = \begin{cases}
0 & \text{for } -\frac{a}{2} \leq x \leq \frac{a}{2}, \\
\infty & \text{otherwise.}
\end{cases}$$
- Relevant Equations
- Schrödinger equation.
We want to solve the Schrödinger equation in the region ##-a/2 \leq x \leq a/2## where ##V(x) = 0##.
The equation is:$$\frac{d^2\psi(x)}{dx^2} + k^2\psi(x) = 0 \quad \text{where} \quad k^2 = \frac{2mE}{\hbar^2}$$ The general solution is:$$\psi(x) = A\sin(kx) + B\cos(kx)$$Applying Boundary Conditions
We know ##\psi(-a/2) = \psi(a/2) = 0##, which gives:$$\begin{cases}
-A\sin(ka/2) + B\cos(ka/2) = 0 \\
A\sin(ka/2) + B\cos(ka/2) = 0
\end{cases} $$
##\textbf{Case 1:}## If ##B = 0## (sine solutions)
The equations reduce to ##A\sin(ka/2) = 0##
Non-trivial solution requires ##\sin(ka/2) = 0##
This gives ##k = \frac{2n\pi}{a}##for ##n=1,2,3,...##
Wave function: ##\psi(x) = A\sin\left(\frac{2n\pi x}{a}\right)##
##\textbf{Case 2:}## If ##\cos(ka/2) = 0## (cosine solutions)
Then ##ka/2 = \frac{n\pi}{2}## for odd ##n=1,3,5,...##
Wave function has both sine and cosine functions ...
Where I'm Stuck:
My book gives different solutions than what I derived here.
For Case 2, It is has only cosine function.
Would appreciate any insight into this discrepancy.
The equation is:$$\frac{d^2\psi(x)}{dx^2} + k^2\psi(x) = 0 \quad \text{where} \quad k^2 = \frac{2mE}{\hbar^2}$$ The general solution is:$$\psi(x) = A\sin(kx) + B\cos(kx)$$Applying Boundary Conditions
We know ##\psi(-a/2) = \psi(a/2) = 0##, which gives:$$\begin{cases}
-A\sin(ka/2) + B\cos(ka/2) = 0 \\
A\sin(ka/2) + B\cos(ka/2) = 0
\end{cases} $$
##\textbf{Case 1:}## If ##B = 0## (sine solutions)
The equations reduce to ##A\sin(ka/2) = 0##
Non-trivial solution requires ##\sin(ka/2) = 0##
This gives ##k = \frac{2n\pi}{a}##for ##n=1,2,3,...##
Wave function: ##\psi(x) = A\sin\left(\frac{2n\pi x}{a}\right)##
##\textbf{Case 2:}## If ##\cos(ka/2) = 0## (cosine solutions)
Then ##ka/2 = \frac{n\pi}{2}## for odd ##n=1,3,5,...##
Wave function has both sine and cosine functions ...
Where I'm Stuck:
My book gives different solutions than what I derived here.
For Case 2, It is has only cosine function.
Would appreciate any insight into this discrepancy.