Consider a particle of mass m subject to a one-dimensional...

Consider a particle of mass $m$ subject to a one-dimensional potential $V(x)$ that is given by $$ V=\infty, \quad x<0 ; \quad V=0, \quad 0 \leqslant x \leqslant a ; \quad V=V_{0}, \quad x>a $$ Show that bound $\left(E<V_{0}\right)$ states of this system exist only if $k \cot k a=-\kappa$ where $k^{2}=2 m E / \hbar^{2}$ and $\kappa^{2}=2 m\left(V_{0}-E\right) / \hbar^{2}$

Consider a particle of mass $m$ subject to a one-dimensional potential $V(x)$ that is given by
$$
V=\infty, \quad x<0 ; \quad V=0, \quad 0 \leqslant x \leqslant a ; \quad V=V_{0}, \quad x>a
$$
Show that bound $\left(E<V_{0}\right)$ states of this system exist only if $k \cot k a=-\kappa$ where $k^{2}=2 m E / \hbar^{2}$ and $\kappa^{2}=2 m\left(V_{0}-E\right) / \hbar^{2}$

Key Concepts

Bound States

Bound states are specific quantum states where the particle's energy is confined below the potential barrier, leading to spatially localized wave functions that are normalizable. In this context, the particle’s energy remains below the threshold V0, resulting in exponential decay of the wavefunction in the classically forbidden region, thereby ensuring that the particle remains confined within a limited region of space.

Time-Independent Schrödinger Equation

The time-independent Schrödinger equation is the fundamental equation that governs the stationary states of a quantum system. It is used to determine the energy eigenstates and associated eigenvalues by solving in different potential regions, leading to solutions that must satisfy the overall energy constraints and normalization conditions of the system.

Boundary Conditions

Boundary conditions are crucial in solving piecewise potential problems as they ensure the continuity of the wavefunction and its derivative at the interfaces between regions with different potentials. The matching of solutions across boundaries, such as the infinite and finite potential regions, leads to quantization conditions and is essential for the physical validity of the solution.

Transcendental Equation

The transcendental equation, derived from applying boundary conditions, is a non-algebraic equation that links the parameters of the system. In this case, the condition k cot(ka) = -? emerges from the continuity requirements at the boundary between the well and the barrier, and it determines the allowed energy eigenvalues for the bound states.

Wave Vector and Decay Constant

The wave vector k and the decay constant ? quantify the oscillatory behavior and exponential decay of the wavefunction, respectively. Specifically, k is related to the kinetic energy of the particle within the region where the potential is zero, while ? characterizes the rate of decay of the wavefunction in the classically forbidden region where the potential exceeds the particle’s energy.

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