Question

For a particle in a one-dimensional box of length L with potential $V(x)=0$ for $0<L>x>0$ and $V(x)=\infty$ otherwise, an acceptable wave function consistent with the boundary conditions is

• A. $A\cos\left(\frac{n\pi x}{L}\right)$
• B. $B(x + x^2)$
• C. $Cx^3(x - L)$
• D. $\dfrac{D}{\sin\left(\frac{n\pi x}{L}\right)}$

Hint:

Wavefunctions in infinite wells must vanish at the boundaries.

Answer 1

The Correct Option is C

Step 1: Boundary conditions.
For an infinite potential well: \[ \psi(0) = 0,\quad \psi(L) = 0 \] Step 2: Test each option.
(A) $\cos(n\pi x/L)$ is not zero at $x=0$. Not allowed.
(B) $x+x^2$ is not zero at $x=0$. Not allowed.
(C) $x^3(x-L)$ is zero at both $x=0$ and $x=L$. Acceptable.
(D) Reciprocal sine diverges; not physical.
Step 3: Conclusion.
Only option (C) satisfies both boundary conditions and is finite everywhere.