Quantum Mechanics 1D harmonic Oscillator

In summary, the conversation is about someone working through exam papers for an upcoming module in Quantum Mechanics. They are struggling with a question that involves the time evolution of a composite state in a 1D harmonic oscillator. They are unsure if the question is asking for the wavefunction for a specific value of n or a general wavefunction. The exact behavior of eigenstates to different energies can be determined from the full Schrodinger equation, but the person believes that an Ansatz involving separation of variables may have been written at the start of solving the harmonic oscillator.
  • #1
dcuk86
4
0
Hi, I'm currently working through some exam papers from previous years before an upcoming module in Quantum Mechanics.

Homework Statement



See the attached image

Homework Equations





The Attempt at a Solution




I'm a little stumped with this one, I'm assuming that I'm looking at a 1D harmonic Oscillator and the wording of the question suggests that the wave function just needs to be stated and not actually proven (?).
In your opinion is this question looking for the wavefunction for n=1 or a general wavefunction in terms of n? its mostly the time dependence which has thrown me.
 

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  • #2
I would say you can take the eigenfunction [tex]u_0,u_1[/tex] as given.
Eigenstates to different energies evolve differently in time.
The exact behaviour can be extracted from the full Schrodinger equation. Although I think you have probably written it at the start of solving the H.O., some Ansatz involving separation of variables.

The question is then looking for the time evolution of this composite state.
 

1. What is a 1D harmonic oscillator in quantum mechanics?

A 1D harmonic oscillator is a simplified model used in quantum mechanics to describe the behavior of a particle in a potential energy well that varies quadratically with position. This potential energy well resembles a spring, hence the name "harmonic oscillator".

2. How does the energy of a 1D harmonic oscillator change with increasing quantum number?

The energy of a 1D harmonic oscillator is quantized, meaning it can only take on certain discrete values. As the quantum number increases, so does the energy of the oscillator. The difference between energy levels is equal to the energy of a single quantum, known as a "quantum of energy".

3. What is the wave function of a 1D harmonic oscillator?

The wave function of a 1D harmonic oscillator is a mathematical function that describes the probability amplitude of finding a particle in a specific position and time. It is a complex-valued function that varies sinusoidally in space and time, with the amplitude decreasing as the particle moves away from the center of the potential well.

4. How does the uncertainty principle apply to a 1D harmonic oscillator?

The uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. In the case of a 1D harmonic oscillator, the uncertainty in position and momentum is related to the uncertainty in the particle's energy. As the energy increases, the uncertainty in position and momentum also increases.

5. How is the 1D harmonic oscillator related to real-world systems?

The 1D harmonic oscillator is a simplified model that can be applied to many real-world systems, such as vibrating molecules, atoms trapped in optical lattices, and the motion of electrons in atoms. It provides a useful framework for understanding the behavior of these systems and making predictions about their properties and behaviors.

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