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The wavefunction of a particle in a box of length $a$ for t = 0 is constant within the range $$\left(\frac{ a}{2}\right) - ϵ$$ and $$\left(\frac{ a}{2}\right) + ϵ$$ and 0 otherwise.

Sketch and normalize this wavefunction.

Where is this particle likely to be found at t = 0?

My question is: is it possible to have a constant wavefunction? The question gives that the wavefunction is constant, but that would mean if you sketch it, it's a horizontal line, and I thought the wave function is always a sinusoidal wave? If it is constant does it still have the formula $$\left(A\sin\frac{ nπx}{a}\right) ?$$ Also if it is constant, isn't it equally likely to be found anywhere within the bounded region at t=0?

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A wavefunction can have basically any shape you like — provided it's bounded and normalizable. The sinusoid you suggest is an eigenfunction of your box (i.e. the wavefunction of a state with definite energy), while the piecewise-constant wavefunction given in the problem is just an initial condition for time-dependent Schrödinger's equation.

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  • $\begingroup$ When it says that the wavefunction is constant though, does it mean that it is constant in space; so that it's shape doesn't change as the wavefunction time evolves? $\endgroup$ – Ella Nov 8 '16 at 13:54
  • $\begingroup$ @Ella depends on the Hamiltonian. With certain Hamiltonians such a wavefunction may be constant in time. But in general such discontinuous wavefunction will immediately spread out in space. See e.g. an animation in this question at Math.SE for approximation of how edge of such wavefunction evolves in constant potential. $\endgroup$ – Ruslan Nov 8 '16 at 13:57

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