Finding the eigenfunction of momentum

In summary, the ground state of an infinite square well is not an eigenfunction of momentum as its derivative does not result in a constant value. The eigenfunction, or wavefunction, is the sine function, while the eigenvalue is the observable (momentum) and the operator is momentum. The term cotangent is not a constant and therefore does not qualify as an eigenvalue. The correct eigenfunction for momentum would be g = exp(i k x) and the corresponding eigenvalue would be hbar k. The terms eigenvector, eigenstate, eigenket, eigenmode, eigenface are all interchangeable depending on the context, but technically "eigenvector" is always correct. The momentum operator is the operator for this system
  • #1
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the questions is: is the ground state of an infinite square well an eigenfunction of momentum, if so what is the momentum?

solution:

DjKMb3q.png


i was working it out and i got something different from the solutions, and i don't understand where they're getting the cotangent term from..


and also, please confirm this for me; the eigenvector is the wavefunction, the eigenvalue is the observable (in this case, the momentum), and the eigenFUNCTION is the operator right?
 
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  • #2
The eigenfunction is the wavefunction (can also be called an eigenvector) and is the sine function here, the eigenvalue would be the observable (a number), and the operator in this case is momentum. If you have an operator P and a function f, f is an eigenfunction of P if P f = p f, where p is some number. If you take the derivative of sine, you don't get a number multiplying sine back; you get the cotangent term, which is not a constant. So sine is not an eigenfunction of P. An example of something that is an eigenfunction of P would be g = exp(i k x) for example. Then P g = hbar k g.
 
  • #3
What is [itex]cot(x)sin(x)[/itex]?

The eigenvalue is the constant you get when you act the operator on a eigenvector/eigenstate/eigenfunction/eigenket/eigenmode/eigenface/etc.

The name you use depends on what is the most precise thing to say. For example, you'd say eigenfunction if you were talking about a wavefunction in the position representation, eigenstate/eigenket if you're still in abstract bra-ket notation, and you'd use the word eigenvector if say for example you were in a linear algebra class working out an eigenvalue equation. But also, the word "eigenvector" is technically always correct in any case since when you solve these types of equations, there is a map to some vector space in the abstract sense.

The operator is the momentum operator
The eigenvector is a momentum eigenstate (which you can write in a specific basis to call it an eigenfunction)
The eigenvalue is the momentum
 

Related to Finding the eigenfunction of momentum

1. What is an eigenfunction of momentum?

An eigenfunction of momentum is a mathematical function that represents the state of a system with a specific momentum value. It is a solution to the eigenvalue equation of the momentum operator in quantum mechanics.

2. Why is finding the eigenfunction of momentum important?

Knowing the eigenfunctions of momentum allows us to determine the momentum values of a system and make predictions about its behavior. It is a fundamental concept in quantum mechanics and is essential for understanding the behavior of particles at the atomic and subatomic level.

3. How is the eigenfunction of momentum calculated?

The eigenfunction of momentum is calculated by solving the eigenvalue equation for the momentum operator. This involves finding the eigenvalues (or possible momentum values) and corresponding eigenfunctions that satisfy the equation.

4. Can the eigenfunction of momentum change over time?

Yes, the eigenfunction of momentum can change over time. In quantum mechanics, the state of a system can change over time, resulting in a change in its eigenfunctions. This is known as the time evolution of a system.

5. What are some real-world applications of the eigenfunction of momentum?

The eigenfunction of momentum has many applications in fields such as quantum mechanics, chemistry, and materials science. It is used to describe the behavior of particles at the atomic and subatomic level, determine the energy levels of atoms and molecules, and study the properties of materials at the nanoscale.

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