• Craptola
In summary, the wave function of a particle is an eigenfunction of the operators for total energy and x component of momentum if the wave function is an exponential. The eigenvalues of the wave function are determined by the given constants k and ω.

#### Craptola

I've been wrestling with this question for a while and can't seem to find anything in my notes that will help me.

## Homework Statement

Determine whether the wave function $\Psi (x,t)= \textrm{exp}(-i(kx+\omega t))$ is an eigenfunction of the operators for total energy and x component of momentum, and if it is, calculate the eigenvalues.

## Homework Equations

Condition for an eigenfunction:
$$\hat{E}\Psi =k\Psi$$
Where K is the eigenvalue
Energy operator:
$$\hat{E}=i\hbar\frac{\partial }{\partial t}$$

## The Attempt at a Solution

Determining that psi is an eigenfunction is easy enough.
$$\hat{E}\Psi =i\hbar\frac{\partial }{\partial t}[\textrm{exp}(-i(kx+\omega t))]$$
$$=-i\hbar i\omega \Psi =\hbar\omega \Psi =\frac{h}{2\pi }2\pi f\Psi =hf\Psi =E\Psi$$

I can't figure out how to calculate the value of E from this information alone. I imagine the same method works for momentum when I figure out what it is.

hmm? You have just calculated the value of E. Now, yeah it is pretty much a similar method to find out if it is also an eigenstate of momentum, once you remember what the operator looks like :)

1 person
Hi Craptola!

hmmm …
Craptola said:
$$\hat{E}\Psi =i\hbar\frac{\partial }{\partial t}[\textrm{exp}(-i(kx+\omega t))]$$
$$=-i\hbar i\omega \Psi =\hbar\omega \Psi$$

stop there?

I assumed that the question wanted me to calculate an actual number for the eigenvalue, which is what confused me as it seems that E could be anything depending on other variables. This is all stuff we covered fairly recently so I wasn't sure if there was some kinda law that wasn't in the lecture notes which limited the possible values of E.

Craptola said:
… it seems that E could be anything depending on other variables.

Ah, but k and ω aren't variables, they're your given constants!

1 person

## 1. What are eigenvalues in quantum mechanics?

Eigenvalues in quantum mechanics refer to the possible values of a physical observable, such as energy or momentum, that can be obtained from a quantum system. They represent the fundamental states of a system and are used to describe the behavior and properties of particles on a microscopic level.

## 2. How are eigenvalues calculated in quantum mechanics?

Eigenvalues in quantum mechanics are calculated using mathematical operators, such as the Hamiltonian, which act on the wavefunction of a particle. The resulting eigenvalues correspond to the possible outcomes of a measurement of the observable associated with that operator.

## 3. What is the significance of eigenvalues in quantum mechanics?

Eigenvalues in quantum mechanics play a crucial role in understanding the behavior of particles at the quantum level. They provide information about the possible states and properties of a system and are used to make predictions about the behavior of particles in experiments.

## 4. Can eigenvalues change in quantum mechanics?

In general, the eigenvalues of a quantum system do not change over time. However, in certain cases, such as when the system is subjected to external forces or interactions, the eigenvalues may change. This can result in changes to the system's behavior and properties.

## 5. How do eigenvalues relate to superposition in quantum mechanics?

In quantum mechanics, a particle can exist in multiple states simultaneously, known as superposition. The eigenvalues of a system correspond to the possible states that a particle can be in, and the coefficients in the wavefunction determine the probability of the particle being in a specific state. Therefore, eigenvalues are closely related to the concept of superposition in quantum mechanics.