Question

Residents of Flatworld-a two-dimensional world far, far away - have it easy. Although quantum mechanics of course applies in their world, the equations they must solve to understand atomic energy levels involve only two dimensions. In particular, the Schrodinger equation for the one-electron flatrogen atom is $$ \begin{aligned} -\frac{\hbar^{2}}{2 m} \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right) & \psi(r, \theta)-\frac{\hbar^{2}}{2 m} \frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}} \psi(r, \theta) \\ &+U(r) \psi(r, \theta)=E \psi(r, \theta) \end{aligned} $$ (a) Separate variables by trying a solution of the form \(\psi(r, \theta)=R(r) \Theta(\theta)\), then dividing by \(R(r) \Theta(\theta)\). Show that the \(\theta\) equation can be written $$ \frac{d^{2}}{d \theta^{2}} \Theta(\theta)=C \Theta(\theta) $$ where \(C\) is a separation constant. (b) To be physically acceptable, \(\Theta(\theta)\) must be continuous, which. since it involves rotation about an axis. means that it must be periodic. What must be the sign of \(C\) ? (c) Show that a complex exponential is an acceptable solution for \(\Theta(\theta)\).

Short answer

By separating variables, the equation \[\frac{d^2}{d \theta^2} \Theta(\theta) = C \Theta(\theta)\] is obtained. \(C\) must be positive to ensure the physical acceptability of the wave function as it involves rotation. A complex exponential function also satisfies the equation and is therefore an acceptable solution.

Step-by-step solution

Separate Variables

The suggested solution is a product of radial and angular components, i.e. \(\psi(r, \theta)=R(r) \Theta(\theta)\). The given equation is divided by \(R(r) \Theta(\theta)\) to separate variables.

Obtain the Angular Part

After dividing by \(R(r) \Theta(\theta)\), the angular part can be isolated on one side of the equation as follows: \[ -\frac{1}{\Theta(\theta)} \frac{\partial^2}{\partial \theta^2} \Theta(\theta) = C \] It gives us the differential equation for \(\Theta(\theta)\) in terms of \(C\), which is the separation constant.

Discuss Physical Acceptability

The solution \(\Theta(\theta)\) must be continuous and periodic as it involves rotation about an axis. This requires \(C\) to be positive because a negative \(C\) might lead to a periodic but non-continuous solution, which is physically unacceptable.

Acceptability of Complex Exponential

Moreover, a complex exponential \(\Theta(\theta) = e^{in\theta}\), where \(n\) is an integer, can also satisfy the above equation for positive \(C\). The squared modulus of complex exponential remains constant during rotation as it is unity, making it an acceptable solution.

Key concepts

Schrodinger Equation

In the world of quantum mechanics, the Schrodinger equation plays a pivotal role as it provides us with crucial information about the atomic energy levels. For Flatworld, which exists in just two dimensions, we have a particular form of this equation to explore. The equation involves both radial and angular parts, and describes the energy and behavior of an electron in a flatrogen atom.

Mathematically, the Schrodinger equation in two dimensions for this problem is given by: \[-\frac{\hbar^{2}}{2 m} \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right) \psi(r, \theta) - \frac{\hbar^{2}}{2 m} \frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}} \psi(r, \theta) + U(r) \psi(r, \theta) = E \psi(r, \theta)\] Here, \( \hbar \) is the reduced Planck's constant, \( m \) is the electron's mass, \( U(r) \) is the potential energy, and \( E \) is the total energy. \( \psi(r, \theta) \) is the wave function that describes the state of the electron. By solving this equation, we can deduce the possible energy levels and behaviors of particles confined to move in this two-dimensional space.

Understanding how to work with this equation is fundamental in quantum mechanics, as it directly relates to how particles behave on an atomic scale.

Variable Separation

To effectively solve the Schrodinger equation in two dimensions, we can employ a method called variable separation. This technique simplifies complex equations by breaking them down into more manageable parts. In our case, we express the wave function as a product of two separate functions: one that depends only on the radial coordinate \( r \) and another that depends only on the angular coordinate \( \theta \).

We start with the expression: \[\psi(r, \theta) = R(r) \Theta(\theta)\] Here, \( R(r) \) is the radial part and \( \Theta(\theta) \) is the angular part. This allows us to handle each coordinate independently. By dividing the Schrodinger equation by \( R(r) \Theta(\theta) \), we isolate parts of the equation to form separate differential equations for \( R \) and \( \Theta \).

The angular part leads to the equation:\[\frac{d^{2}}{d \theta^{2}} \Theta(\theta) = C \Theta(\theta)\] where \( C \) is a constant from the separation process. Handling complex equations becomes much easier after separating the variables, revealing useful insights about the quantum system in question.

Periodic Solutions

Periodic solutions are essential when dealing with the angular component \( \Theta(\theta) \) of the Schrodinger equation in polar coordinates. Given that \( \theta \) corresponds to rotation, any physical solution \( \Theta(\theta) \) must be continuous across the range of \( \theta \) from 0 to \( 2\pi \) and further. This implies that \( \Theta(\theta) \) must be periodic.

A classic form of periodic function is the complex exponential function. It turns out that \( \Theta(\theta) = e^{in\theta} \), with \( n \) being an integer, solves the angular equation provided that \( C \), the separation constant, is positive. \( C \) must be positive because if it were negative, the solution might become non-continuous, which conflicts with the physical requirements of periodicity.

Using complex exponentials, periodicity is ensured because as \( \theta \) goes from 0 to \( 2\pi \), the function returns to its initial value, forming a seamless cycle. Such periodic solutions not only satisfy the mathematical requirements but also align well with the physical interpretation of rotations in flatworld's quantum setting.