# Proving time independence of energy eigenstates

• Gnator
In summary, the conversation is about finding the probability density of psi(x,t) and the confusion surrounding the use of the exponential term. The solution is to multiply psi by its complex conjugate instead of squaring it.
Gnator

## Homework Statement

We are ask to find the probability density of psi(x,t). I know that psi have an exp term but i don't understand how by squaring psi make the exp term disappear.

## Homework Equations

Psi = sqrt(2/L)sin(n*pi*x / L)e^(-2*pi* i(E/h)t

## The Attempt at a Solution

I attempted changing the exp term into polar coord but I can't seem to get anywhere with that.

You don't square psi. You multiply it by it's complex conjugate. Makes all the difference.

Yes, I just read my text and I discover you are suppose to multiple the complex conjugate. Thank you.

## 1. What is the concept of time independence in energy eigenstates?

Time independence in energy eigenstates refers to the property of a physical system in which the energy eigenstates do not change over time. This means that the energy of the system remains constant, regardless of the time at which it is measured.

## 2. How do we prove time independence of energy eigenstates?

To prove time independence of energy eigenstates, we use the time-independent Schrödinger equation, which states that the energy eigenstates are the solutions to the equation Hψ = Eψ, where H is the Hamiltonian operator, ψ is the wave function, and E is the energy of the system. If the solutions to this equation do not change over time, then the energy eigenstates are time independent.

## 3. Why is it important to prove time independence of energy eigenstates?

Proving time independence of energy eigenstates is important because it allows us to accurately predict the behavior of physical systems. If the energy eigenstates are not time independent, then the energy of the system will change over time, making it difficult to make accurate predictions about the system's behavior.

## 4. What are some real-world examples of time-independent energy eigenstates?

One example of time-independent energy eigenstates is the hydrogen atom. The energy levels of the hydrogen atom remain constant over time, making it possible to accurately predict the behavior of the atom. Another example is a simple pendulum, where the energy of the pendulum remains constant as it swings back and forth.

## 5. Are all energy eigenstates time independent?

No, not all energy eigenstates are time independent. In some cases, the energy eigenstates may change over time due to external influences or interactions with other systems. However, in many cases, such as in simple systems like the hydrogen atom, the energy eigenstates are time independent.

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