Quantum Harmonic Oscillator

This means that the state |psi> does not change direction abruptly, which is a characteristic of a continuous function.In summary, we can show that |psi> is a continuous function of time by using the fact that the expectation value of the Hamiltonian is constant in time and applying the Ehrenfest theorem. This means that the state |psi> does not change direction abruptly, which is a characteristic of a continuous function. As for the second part of the problem, I'm still working on that. Let me know if you have any thoughts on it!
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Thanks in advance for reading/replying

Homework Statement


In the interval (t+dt, t) the Hamiltonian H of some system varies in such a way that |H|psi>| remains finite. Show that under these circumstances |psi> is a continuous function of time.

A harmonic oscillator with frequency w is in its ground state when the stiffness of the spring is instantaneously reduced by a factor (f^4)<1 such that its natural frequency becomes (f^2)w. What is the probability that the oscillator is subsequently found to have energy 1.5.(h/2pi).w.f^2?


Homework Equations


En = (n + 0.5)(h/2pi)w


The Attempt at a Solution



For the first part I thought maybe I needed to use ehrenfest, as |H|psi>|^2 = <psi|H|psi>. So then I think you get

d/dt <psi|H|psi> = <psi|dH/dt|psi>

However I don't really know what to do with this and am probably way off.

For the second part I'm really confused. The energy it talks about is the energy eigenvalue of the first excited state; but I don't know how to get a probability without the state |psi> being given to me. Initially the state is the ground stationary state for the original frequency w, so initially a measurement will definitely find the energy value (h/2pi)w. So the state must change, but how do I work out what it changes to?


Argh
 
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  • #2
, I am stuck on this problem too! It seems like we are both trying to use the Ehrenfest theorem to solve this problem, but I'm not sure if that's the right approach. Let me try to break down the problem and see if we can make some progress.

First, the problem states that in the interval (t+dt, t), the Hamiltonian H varies in such a way that |H|psi>| remains finite. This means that the Hamiltonian is changing in time, but the state |psi> does not become infinite or undefined at any point during this interval. This suggests that the Hamiltonian is changing smoothly and continuously, which would lead to the conclusion that |psi> is also a continuous function of time.

So how do we prove this? Well, we can start by using the Ehrenfest theorem, which states:

d/dt <A> = (1/i*h)<[H,A]>

where <A> is the expectation value of the observable A, [H,A] is the commutator of the Hamiltonian and the observable A, and i is the imaginary unit. In this case, we are interested in the expectation value of the Hamiltonian itself, so we have:

d/dt <H> = (1/i*h)<[H,H]>

But the commutator of a quantity with itself is always zero, so this simplifies to:

d/dt <H> = 0

This means that the expectation value of the Hamiltonian is constant in time. Now, we can use this fact to show that |psi> is also a continuous function of time. We know that the expectation value of the Hamiltonian is given by:

<H> = <psi|H|psi>

Since this is constant in time, it must also be true that:

d/dt <psi|H|psi> = 0

But we also know that:

d/dt <psi|H|psi> = <psi|dH/dt|psi> + <psi|H|dpsi/dt>

Since the first term is zero, this means that:

<psi|H|dpsi/dt> = 0

This tells us that the inner product <psi|H|dpsi/dt> is zero, which in turn means that the vector |dpsi/dt> is orthogonal to the vector <psi|H|. In other words, |dpsi/dt
 

What is a Quantum Harmonic Oscillator?

A Quantum Harmonic Oscillator is a physical system that exhibits oscillatory motion, similar to a pendulum or a mass on a spring, but at a quantum level. It is a fundamental concept in quantum mechanics and is used to describe the behavior of atoms, molecules, and other quantum systems.

What are the main characteristics of a Quantum Harmonic Oscillator?

The main characteristics of a Quantum Harmonic Oscillator include its energy levels, which are quantized and evenly spaced, its wave function, which follows the Gaussian distribution, and its potential energy, which is parabolic in shape. It also exhibits properties such as zero-point energy and uncertainty in position and momentum.

What is the significance of the Quantum Harmonic Oscillator in quantum mechanics?

The Quantum Harmonic Oscillator is a fundamental concept in quantum mechanics and is used to model and understand various physical systems at a quantum level. It allows for the quantization of energy levels and provides a framework for understanding the behavior of atoms, molecules, and other quantum systems. It also serves as a building block for more complex quantum systems and has applications in fields such as quantum computing and quantum optics.

How is the Quantum Harmonic Oscillator described mathematically?

The Quantum Harmonic Oscillator is described by the Schrödinger equation, which is a differential equation that relates the wave function of the system to its energy and potential. The harmonic oscillator potential is represented by a parabolic function, and the wave function is typically expressed as a Gaussian distribution. The energy levels of the system are quantized and can be calculated using the Schrödinger equation.

What are some real-life examples of Quantum Harmonic Oscillators?

Some real-life examples of Quantum Harmonic Oscillators include vibrating molecules, such as diatomic molecules like CO or HCl, and vibrating atoms in a crystal lattice. The motion of electrons in an atom can also be described as a quantum harmonic oscillator. Additionally, some macroscopic systems, such as superconducting circuits, can exhibit quantum harmonic oscillator behavior under certain conditions.

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