# Getting Psi(x,t) from Psi(x,0)

• neo2478
In summary, the conversation is about a problem involving a particle in an infinite square well and finding its wave function at a specific time. The initial wave function is given and needs to be normalized before plugging it into Schrödinger's equation. The conversation also discusses the use of summation instead of integration for discrete energies and the importance of finding the coefficients using orthonormality.

#### neo2478

Hey guys, first time poster.

I am doing some quantum physics homework, and I came across the following problem:

A particle in an infinite square well has the initial wave function
Psi(x,0) = Ax 0?x?a/2
A(a-x) a/2 ?x?a

Find Psi (x,t)

Now after normalizing it, I tried plugging it in Schrödinger's equation, however I'm still having problems.

Psi(x,t)=Sum exp{-iE_n hbar t}phi_n(x)<phi_n|Psi(x,0)>,
where phi_n are the estates and E_n the eignevalues of the square well.

Why did you do the summation of the function instead of integrating it??

neo2478 said:
Why did you do the summation of the function instead of integrating it??

The energies are discrete, hence the summation.

You must write
$$\Psi(x,t=0) = \sum_n c_n \psi_n(x)$$
where the $\psi_n(x)$ are the energy eigenstates, $\sqrt{2/a} \, sin(n \pi x/a)$ (for a well located between x=0 and x=a). What you have to do is to find the coefficients c_n using the orthonormality of the sine wavefunctions. Once you have that, the wavefunction at any time is

$$\Psi(x,t) = \sum_n c_n e^{-i E_n t / \hbar} \psi_n(x)$$

## 1. How do you get Psi(x,t) from Psi(x,0)?

To get Psi(x,t) from Psi(x,0), you need to use the Schrodinger equation, which is a mathematical equation that describes the time evolution of a quantum system. This equation involves the Hamiltonian operator, which operates on the wave function Psi(x,t) to give the time derivative of the wave function. By solving this equation, you can obtain the wave function Psi(x,t) at any given time t.

## 2. What is the significance of Psi(x,t) in quantum mechanics?

Psi(x,t) is a fundamental concept in quantum mechanics, as it represents the probability amplitude of a quantum system at a given time t. It contains all the information about the state of the system and can be used to calculate the probability of finding the system in a particular state. In other words, the square of Psi(x,t) gives the probability of finding the system in the state represented by the wave function.

## 3. Can Psi(x,t) be obtained experimentally?

No, Psi(x,t) cannot be directly measured or observed experimentally. It is a mathematical construct that represents the quantum state of a system and can only be calculated or inferred through measurements of observables such as position or momentum. However, the predictions and behaviors of Psi(x,t) have been confirmed through various experiments in quantum mechanics.

## 4. How does Psi(x,t) relate to the uncertainty principle?

The uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. This is because the wave function Psi(x,t) contains information about both position and momentum, and the more accurately one of these properties is known, the less accurately the other can be known. Therefore, the uncertainty principle arises from the fundamental nature of Psi(x,t) in quantum mechanics.

## 5. Can Psi(x,t) be used to predict future states of a quantum system?

Yes, the Schrodinger equation allows for the calculation of Psi(x,t) at any given time t, which can then be used to predict the future state of the system. However, this prediction is probabilistic in nature and does not guarantee a specific outcome. The wave function can also be affected by external factors, making it impossible to accurately predict the exact state of a quantum system at a future time.