Solving Psi(x,t) for a Harmonic Oscillator

• Talib
In summary, Psi(x,t) is a complex-valued wave function that describes the quantum state of a particle in a harmonic oscillator potential. It can be solved using the time-dependent Schrodinger equation and provides information about the probability of finding the particle at a certain position and time. While it can be solved analytically for all harmonic oscillator systems, it only applies to non-relativistic systems and assumes a purely quadratic potential energy function.
Talib
Hello,

A harmonic oscillator is in the initial state:
Psi(x, 0) = Phi_n (x)
where Phi_n(x) is the nth solution of the time-independent Schr¨odinger equation.
What is Psi(x, t)?

Any clue?

Thanks

Psi(x, t) is the solution of the time dependant Schrödinger equation (TDSE). The TDSE is a separable partial differential equation. More precisely, the solution to the TDSE is the product of the TISE and of [itex]\exp(iEt/\hbar)[/tex]:

$$\Psi(x,t)=\psi(x)e^{iEt/\hbar}$$

Thanks a lot :):)

1. What is Psi(x,t) in the context of a harmonic oscillator?

Psi(x,t) is the wave function that describes the quantum state of a particle in a harmonic oscillator potential. It is a complex-valued function that gives the probability amplitude for finding the particle at a certain position (x) and time (t).

2. How do you solve for Psi(x,t) in a harmonic oscillator system?

To solve for Psi(x,t) in a harmonic oscillator, you can use the time-dependent Schrodinger equation and apply the appropriate boundary conditions. This will yield a solution in the form of a wave function that describes the quantum state of the particle.

3. What is the significance of solving for Psi(x,t) in a harmonic oscillator?

Solving for Psi(x,t) in a harmonic oscillator allows us to understand the quantum behavior of a particle in a harmonic potential. It provides information about the probability of finding the particle at a certain position and time, and can also be used to calculate other physical properties of the system.

4. Can Psi(x,t) be solved analytically for all harmonic oscillator systems?

Yes, Psi(x,t) can be solved analytically for all harmonic oscillator systems. This is because the potential energy function of a harmonic oscillator is quadratic, which makes the Schrodinger equation solvable using standard mathematical techniques.

5. Are there any limitations to using Psi(x,t) to describe a harmonic oscillator system?

One limitation of using Psi(x,t) to describe a harmonic oscillator system is that it only applies to non-relativistic systems. Additionally, it assumes that the potential energy function is purely quadratic, which may not always be the case in real-world systems.

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