Sketch the wave function. Is it smooth?

$\mathbf{\psi }\left(x\right)\mathbf{=}\{\begin{array}{l}2\sqrt{a^3{\mathrm{xe}}^{-\mathrm{ax}}}X>0\\ 0X<0\end{array}$

In quantum physics, a wave function is a variable number that mathematically characterises a particle's wave properties. The value of a particle's wave function at a given place in space and time is proportional to the probability of the particle being there at that time.

Description of motion of a particle in quantum mechanics is described by wave function, which is a function of both position and time. The wave function is continuous in nature and it satisfied the Schrödinger equation.

Write the expression of the wave function of a particle for x > 0.

$\mathrm{\Psi }\left(\mathrm{x}\right)=2\sqrt{{\mathrm{a}}^3}{\mathrm{xe}}^{-\mathrm{ax}}$

Write the expression of the wave function of a particle for x < 0.

$\mathrm{\Psi }\left(\mathrm{x}\right)=0$

Draw a diagram to show the variation of $\mathrm{\psi }\left(\mathrm{x}\right)$ with x.

Draw a diagram to show the variation of $\mathrm{\psi }\left(\mathrm{x}\right)$with x.

As the potential energy is infinite for infinite well, so the derivative of the wave function is discontinuous. Thus the curve for variation of $\mathrm{\Psi }\left(\mathrm{x}\right)$with x has positive slope at right and negative slope at left.

Thus, the diagram for the variation of $\mathrm{\Psi }\left(\mathrm{x}\right)$with x is given above.