# Solving a Piecewise Schrodinger equation

• cc94
In summary, the conversation discusses the attempt to solve the time-independent Schrodinger's equation for a specific well, with a focus on cases where the energy of a particle is less than the potential energy. The conversation also touches on the use of exponential solutions and boundary conditions.
cc94

## Homework Statement

I was trying to solve the time-independent Schrodinger's equation for this well: http://i.imgur.com/C9QrvkX.png
First I tried to look at cases where the energy of a particle is ##E < V_1##.

## Homework Equations

Schrodinger's equation:
$$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi(x) - V(x)\psi(x) = E\psi(x)$$

## The Attempt at a Solution

I thought the normal way to deal with these square barriers is to break the equation up piece-wise and then match the conditions at the boundary. But in region III, there's a problem. Schrodinger's equation requires an exponential solution because ##V_1>E##. The wave function is

$$\psi_{III} (x) = Fe^{-\mu x} + Ge^{\mu x},$$

where ##\mu = \frac{\sqrt{2m(V_1-E)}}{\hbar}##. But the boundary condition requires that at ##x = a_3##, ##\psi_{III} (a_3) = 0##. This isn't possible unless either F = G = 0, or the wave function is actually a sin/cos wave function here. What am I missing?

cc94 said:
But the boundary condition requires that at ##x = a_3##, ##\psi_{III} (a_3) = 0##. This isn't possible unless either F = G = 0, or the wave function is actually a sin/cos wave function here. What am I missing?
Are you sure? How about ##F=\exp \mu a_3## and ## G=-\exp -\mu a_3##?

Doh, I didn't think hard enough. Thanks

## 1. What is a piecewise Schrodinger equation?

A piecewise Schrodinger equation is a mathematical equation used in quantum mechanics to describe the behavior of a quantum system. It is a combination of two or more Schrodinger equations, each applicable to a different region of space or time.

## 2. How is a piecewise Schrodinger equation solved?

A piecewise Schrodinger equation is typically solved using a technique called separation of variables, where the equation is split into simpler equations that can be solved individually. The solutions from each region are then matched at the boundaries to form a complete solution.

## 3. What are the applications of solving a piecewise Schrodinger equation?

Solving a piecewise Schrodinger equation is essential in understanding the behavior of quantum systems, such as atoms, molecules, and subatomic particles. It is also used in various fields such as chemistry, material science, and quantum computing.

## 4. What are the challenges in solving a piecewise Schrodinger equation?

The main challenge in solving a piecewise Schrodinger equation is accurately defining the boundaries between different regions. These boundaries can be complex and require advanced mathematical techniques to properly account for them in the equation.

## 5. Are there any alternative methods for solving a piecewise Schrodinger equation?

Yes, there are alternative methods such as numerical methods, which use computers to approximate the solution. However, these methods may not be as accurate as the analytical solution obtained through separation of variables, especially for complex systems.

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