-?? (Section 7.8) Consider the problem of a particle in...

-?? (Section 7.8) Consider the problem of a particle in finite square well of depth $U_{0}$, as described in Section 7.8. To solve this problem, use a coordinate system centered on the box, with the left edge of the well at $x=-a / 2$ and the right edge at $x=+a / 2 .$ From the symmetry of the potential, one can argue that the stationary states should have symmetric probability distributions, that is, $|\psi(x)|^{2}=|\psi(-x)|^{2}$. This implies that solutions are either symmetric and satisfy $\psi(x)=\psi(-x)$ or are antisymmetric and satisfy $\psi(x)=-\psi(-x)$. Thus, with our choice of coordinates, the solutions within the well are either of the form $\cos (k x)$ (for $n=1,3,5, \ldots)$ or of the form $\sin (k x)$ (for $n=2,4,6, \ldots)$. (a) For the case of the symmetric solutions, derive an equation relating $k$ and $\alpha$. [Hint: Using the boundary conditions that both $\psi(x)$ and $\psi^{\prime}(x)$ are continuous at $x=-a / 2$, you can produce two equations that relate $k, \alpha$, and the arbitrary coefficients $A$ and $G$ in Section 7.8. By dividing these equations, you can eliminate the coefficients. The final equation you produce is a transcendental equation relating $E$ and $U_{0}$; it cannot be solved for $E$ using elementary methods.] (b) Show that the equation derived in (a) produces the correct stationary-state energies in the limit $U_{0} \rightarrow \infty$. (c) Using numerical techniques, find the ground-state energy $E_{1}$ of a square well of depth $U_{0}=3 E_{1} .[$ Hint $:$ A simple trialand-error search for the solution works surprisingly well.]

-?? (Section 7.8) Consider the problem of a particle in finite square well of depth $U_{0}$, as described in Section 7.8. To solve this problem, use a coordinate system centered on the box, with the left edge of the well at $x=-a / 2$ and the right edge at $x=+a / 2 .$ From the symmetry of the potential, one can argue that the stationary states should have symmetric probability distributions, that is, $|\psi(x)|^{2}=|\psi(-x)|^{2}$. This implies that solutions are either symmetric and satisfy $\psi(x)=\psi(-x)$ or are antisymmetric and satisfy $\psi(x)=-\psi(-x)$. Thus, with our choice of coordinates, the solutions within the well are either of the form $\cos (k x)$ (for $n=1,3,5, \ldots)$ or of the form $\sin (k x)$
(for $n=2,4,6, \ldots)$. (a) For the case of the symmetric solutions, derive an equation relating $k$ and $\alpha$. [Hint:
Using the boundary conditions that both $\psi(x)$ and $\psi^{\prime}(x)$ are continuous at $x=-a / 2$, you can produce two equations that relate $k, \alpha$, and the arbitrary coefficients $A$ and $G$ in Section 7.8. By dividing these equations, you can eliminate the coefficients. The final equation you produce is a transcendental equation relating $E$ and $U_{0}$; it cannot be solved for $E$ using elementary methods.] (b) Show that the equation derived in (a) produces the correct stationary-state energies in the limit $U_{0} \rightarrow \infty$. (c) Using numerical techniques, find the ground-state energy $E_{1}$ of a square well of depth $U_{0}=3 E_{1} .[$ Hint $:$ A simple trialand-error search for the solution works surprisingly well.]

Key Concepts

Numerical Methods in Quantum Mechanics

Numerical techniques, such as trial-and-error, bisection, or more sophisticated root-finding algorithms, are often employed to solve the transcendental equations that arise in problems like the finite square well. These methods are essential for determining the energy eigenvalues when analytical solutions are not feasible. Numerical approaches also allow for the exploration of parameter spaces and the behavior of quantum systems under various potential configurations.

Infinite Well Limit

In the limit where the potential depth approaches infinity, the finite square well model simplifies to the infinite square well. This idealized case has well-known analytic solutions, making it a useful benchmark for verifying more general results. Demonstrating that the transcendental equations of the finite well reduce to the correct energy eigenvalues in the infinite well limit provides an important consistency check for the model.

Transcendental Equations

Transcendental equations arise when applying the boundary conditions to systems with piecewise constant potentials, such as the finite square well. These equations typically cannot be solved analytically using elementary methods and require numerical or graphical techniques to find the roots. They link the parameters of the problem, like the energy of the state and the depth of the well, and are instrumental in determining the quantized energy levels.

Finite Square Well

The finite square well is a quantum mechanical model where a particle is confined within a potential well of finite depth. Unlike the infinite square well, the potential outside the well is not infinitely high, which allows for the phenomenon of tunneling and leads to both bound (discrete energy levels) and scattering states. The analysis of this system involves solving the Schrödinger equation with piecewise constant potential regions and matching solutions at the boundaries of the well.

Symmetry in Quantum Mechanics

Symmetry considerations in quantum mechanics help determine the properties of the wave functions. In potentials that are symmetric about a point, the stationary states can be classified into symmetric (even) and antisymmetric (odd) solutions. This classification reduces the complexity of the problem by allowing the use of cosine functions for even states and sine functions for odd states, which are automatically consistent with the mirror symmetry of the potential.

Boundary Conditions

In solving quantum mechanical problems like the finite square well, the continuity of the wave function and its first derivative at the boundaries is crucial. These boundary conditions ensure that the Schrödinger equation is satisfied across potential discontinuities, leading to equations that relate the parameters (such as the wave number k inside the well and the decay constant outside) and ultimately determine the allowed energy levels of the system.

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