Question

A spherical infinite well has potential energy $$ U(r)=\left\\{\begin{array}{ll} 0 & ra \end{array}\right. $$ Since this is a central force, we may use the Schrödinger equation in the form \((7-30)-\) - that is, just before the specific hydrogen atom potential energy is inserted. Show that the following is a solution $$ R(r)=\frac{A \sin b r}{r} $$ Now apply the appropriate boundary conditions, and in so doing, find the allowed angular momenta and energies for solutions of this form.

Short answer

The angular momenta are quantized as \(\hbar\sqrt{l(l+1)}\) where l = 0, 1, 2,..., n-1 and the allowed energies of this system are \(E = \frac{\hbar^{2}\pi^{2}n^{2}}{2ma^{2}}\) where n is a positive integer.

Step-by-step solution

Solving the Radial Schrödinger Equation

Firstly, we have to solve the radial Schrödinger equation for this system. This equation does not have a potential energy function U(r) for \(r < a\) and it approximates to infinite for \(r > a\). Therefore, the radial part of the Schrödinger equation simplifies to \(\frac{d^{2}u}{dr^{2}} = -\frac{2mE}{\hbar^{2}}u\) where u(r) = rR(r) is the radial part of the wave function and E is the energy of the system. Here A and b are constants to be determined.

Substituting the Given Wave Function

Substitute the given solution of the wave function \(R(r) = \frac{A sin(br)}{r}\) into the radial Schrödinger equation. This will give the equation \(-b^{2} = -\frac{2mE}{\hbar^{2}}\). Solving for E, we get \(E = \frac{\hbar^{2}b^{2}}{2m}\).

Applying Boundary Conditions

The boundary conditions are: (i) The wave function must be finite at r=0. Since sine function is zero at r=0, this condition is met. (ii) The wave function must be zero at r=a (the edge of the potential well), i.e. \(R(a) = 0\). Therefore \(A \frac{\sin(ba)}{a} = 0\). We know A ≠ 0, so this means that \(ba = n\pi\), \(n = 1, 2, 3,...\), giving us the allowed values of b as \(\frac{n\pi}{a}\). Substituting this value of b in the energy equation gives us the allowed energies \(E = \frac{\hbar^{2}\pi^{2}n^{2}}{2ma^{2}}\).

Key concepts

Schrödinger Equation

The Schrödinger equation is a fundamental concept in quantum mechanics, serving as the basis for describing how quantum systems evolve over time. This equation is akin to Newton's laws in classical mechanics, except it applies to the microscopic world. The Schrödinger equation comes in two forms: the time-dependent and the time-independent. For bound systems like the spherical infinite well, the time-independent form is often used.

In our case, the radial Schrödinger equation describes how the quantum state of a particle changes with respect to its position inside the potential well, which is a region where the particle is confined. For regions where the potential energy (\(U(r)\)) is zero, the equation simplifies significantly, allowing us to solve for the wave function (\(R(r)\)), which describes the radial part of the particle's quantum state. It is critical to understand that the allowed energies will have specific values, which leads us to the concept of energy quantization.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of matter and light on the atomic and subatomic scales. It provides a mathematical framework for much of modern physics and chemistry. Quantum mechanics introduces several revolutionary concepts, including wave-particle duality, uncertainty principle, and quantized energy states.

Central to quantum mechanics is the wave function - a mathematical description of the quantum state of a system. The absolutesquared of the wave function gives the probability distribution for a particle's position, momentum, or other physical properties. The wave function and its properties are governed by the Schrödinger equation, and its interpretation is a matter of ongoing debate among physicists.

Wave Function

The wave function in quantum mechanics is denoted by \(\psi\) or \(\Psi\) and encapsulates everything about the state of a quantum system. In the context of the spherical infinite well, the radial wave function \(R(r)\) represents the radial part of the particle's state and tells us how the quantum state behaves within the well. It's crucial that the wave function be properly normalized and respect the established boundary conditions.

The given wave function solution for the spherical infinite well problem, \(R(r)\), incorporates an 'A' term that can be determined by normalization and a 'b' term related to the particle's energy. These parameters are finely tuned by the boundary conditions and embody the principle that not all states are permissible – only those that meet the physical constraints of the system.

Boundary Conditions

Boundary conditions in quantum mechanics play a decisive role in determining the permissible wave functions for a quantum system. For a particle in a spherical infinite well, the wave function must satisfy certain conditions. Firstly, the wave function must be finite everywhere in space – that means at the origin (\(r=0\)), the function cannot be infinite or undefined. This requirement ensures that the probability of finding the particle is finite and physically meaningful.

The second condition for our spherical well is that the wave function must go to zero at the edge of the well (\(r=a\)). This models the infinite potential barrier at that boundary – the particle cannot exist outside the well, so the probability of finding it there must be zero. By applying these conditions, we determine the allowed values for the parameters in our wave function, which subsequently lead to the quantization of angular momentum and energy.

Energy Quantization

Energy quantization is the concept that energy can only have certain discrete values in a quantum system, contrasting with classical physics where energy is often assumed to be continuous. This phenomenon is one of the cornerstones of quantum mechanics and is observable in multiple areas, such as the discrete emission spectra of atoms and energy levels of electrons in an atom.

In the infinite spherical well problem, energy quantization emerges from applying the boundary conditions to the Schrödinger equation. Specifically, the conditions dictate that the wave function must be zero at the boundary (\(r=a\)), leading to discrete allowable values for the variable 'b', which in turn translate into discrete energy levels. The energy level of the system can only be one of these values, something that has profound implications for the behavior of quantum systems and forms the underpinning of quantum chemistry and electronics.