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In the ideal setup, the wavefunction should be zero at the walls because these walls are of infinite potential, i.e. the particle cannot be in that position of such a high potential (I am trying to understand the relation between the waveshape and the actual physical situation).

But why do figures always depict the time-invariant wavefunction in a sinusoidal form as can be seen on this hyperphysics page? Aren't other irregular or non-symmetric shapes allowed as long as they satisfy the boundary conditions? If no, why is that?

Xfce4
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1 Answers1

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You are right; the wave function does not need to be sinusoidal for the particle in a box. We can have other functions that satisfy the boundary conditions.

The reason so much focus is given to the sine waves is because those functions are wave functions for particles with definite energy. Furthermore, these functions serve as a basis to describe any other wave function. If $\psi_E(x)$ is the wave function that satisfies the Time Independent Schrodinger Equation for a particle with energy $E$ (one of those sine waves), then any wave function $\Psi(x,t)$ can be written as $$\Psi(x,t)=\sum_Ec_Ee^{-iEt/\hbar}\psi_E(x)$$

where $c_E$ are complex constants. Notice that, while $\Psi$ can be written as a sum of sine waves, $\Psi$ itself does not need to be a sine wave.

BioPhysicist
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