A particle of mass m is bound in a square well of width W...

A particle of mass $m$ is bound in a square well of width $W$ and height $P_{0}$. The eigenfunction for the ground state is given by $$\psi(x)=B \cos \left(\frac{4 \pi x}{5 W}\right)$$ for $-W / 2 \leq x \leq W / 2$ and by $\psi=A e^{-|a x|}$ outside the well. (a) Taking $m$ and $\mathbb{W}$ as given, find $\alpha, E$ and $P_{0}$. Compare the value of $E_{1}$ with that of the infinite square well. (b) Find $A$ in terms of $B$ and the given quantities. (c) Set up the integral from which $B$ can be calculated.

A particle of mass $m$ is bound in a square well of width $W$ and height
$P_{0}$. The eigenfunction for the ground state is given by
$$\psi(x)=B \cos \left(\frac{4 \pi x}{5 W}\right)$$
for $-W / 2 \leq x \leq W / 2$ and by $\psi=A e^{-|a x|}$ outside the well.
(a) Taking $m$ and $\mathbb{W}$ as given, find $\alpha, E$ and $P_{0}$. Compare the value of $E_{1}$ with that of the infinite square well.
(b) Find $A$ in terms of $B$ and the given quantities.
(c) Set up the integral from which $B$ can be calculated.

Key Concepts

Normalization of Wavefunctions

Normalization is a fundamental principle in quantum mechanics which ensures that the total probability of finding the particle somewhere in space is one. The coefficients in the wavefunctions (like those inside and outside the well) are determined by imposing this normalization condition through integrating the square of the absolute value of the wavefunction over all space.

Boundary and Continuity Conditions

The solutions to the Schrödinger equation must be continuous and smooth (i.e., their first derivatives must be continuous) at the boundaries between regions with different potential energies. This requirement provides the necessary equations to match the internal and external solutions of the wavefunction, which in turn leads to the quantization of energy levels.

Energy Quantization and Eigenvalue Problems

The need to match the wavefunctions and their derivatives at the potential boundaries results in a discrete set of allowed energy levels for the bound states. Solving the resulting equations shows the quantization of energy – only certain energies are permitted for stable, bound states.

Square Well Potential

This is a fundamental model in quantum mechanics where a particle is confined to a finite region in space by a potential energy function that is lower inside the region and higher outside. The finite square well demonstrates how bound states arise when a particle's energy is less than the potential energy outside the well, leading to exponentially decaying wavefunctions in the classically forbidden region.

Bound State Wavefunctions

In problems involving bound states, the eigenfunctions of the Hamiltonian must satisfy the Schrödinger equation with different forms in the well (where the potential is constant) and outside the well (where the potential is higher). These solutions typically involve sinusoidal functions inside the well and decaying exponential functions outside, ensuring that the wavefunction remains normalizable.

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