Energy eigenstates on a generic symmetric potential. Shooting method

TL;DR

This content discusses the concept of energy eigenstates in quantum mechanics and the shooting method for finding these states.

Transcript

PROFESSOR: Here is your potential. It's going to be a smooth, nice potential like that. V of x. x, x. And now, suppose you don't know anything about the energy eigenstates. Now, this potential will be assumed to be symmetric. So here is one thing you can do. You can exploit some things that you know about this potential. And here's the wave functio... Read More

Key Insights

• 👋 Energy eigenstates in quantum mechanics correspond to specific energy values and are described by wave functions.
• 👋 The behavior of the wave function in the forbidden region and at the boundaries determines the form of the energy eigenstate.
• 👋 The shooting method is a numerical technique used to find energy eigenstates by iteratively adjusting the energy value until a normalizable wave function is obtained.
• 💁 The shooting method requires cleaning up the equation and converting it into dimensionless form before solving it numerically.

Questions & Answers

Q: What are energy eigenstates in quantum mechanics?

Energy eigenstates are the possible states of a quantum system that correspond to specific energy values. They are represented by wave functions that satisfy the Schrodinger equation for a given potential.

Q: How should the wave function of an energy eigenstate behave on the right and left of a potential?

On the right of the potential, the wave function should match the forbidden region wave function. On the left, it should decay, but the specific form of decay depends on the behavior in the middle of the potential.

Q: What is the shooting method in quantum mechanics?

The shooting method is a numerical approach for finding energy eigenstates. It involves iteratively adjusting the energy value and solving the Schrodinger equation until a normalizable wave function is obtained.

Q: How does the shooting method work for finding energy eigenstates?

In the shooting method, an initial energy value is chosen, and the Schrodinger equation is solved with boundary conditions on the wave function and its derivative. By adjusting the energy value and observing the behavior of the wave function, a suitable energy eigenstate can be found.

Summary & Key Takeaways

• The content explains the concept of energy eigenstates and their behavior in a given potential.

• It discusses how the wave function of energy eigenstates should match the forbidden region wave function on the right and decay on the left.

• The content also introduces the shooting method, which is a numerical approach for finding energy eigenstates by iteratively adjusting the energy value until a normalizable wave function is obtained.