# What is the physical meaning for a particle in harmonic oscillator ?

1. Oct 19, 2013

### Outrageous

For infinite square well, ψ(x) square is the probability to find a particle inside the square well.
For hamornic oscillator, is that meant the particle behave like a spring? Why do we put the potential as 1/2 k(wx)^2 ?

Thanks

2. Oct 19, 2013

### Simon Bridge

It means that the potential function has the same maths as that for an ideal harmonic oscillator.
All you need for one of those is a restoring force that depends on the displacement from some equilibrium position.

The 1D HO potential is written $V(x)=\frac{1}{2}\omega^2x^2$ because that is what "Harmonic Oscillator Potential" means. You'll see why it's useful as you study it more.

Some notes:

For an infinite square well, the probability of finding the particle inside the well is 1.

The wavefunction does not give the probability of anything.

3. Oct 19, 2013

### Outrageous

The potential here is the surrounding of the particle? It is a condition for finding the wavefunction when the particle is in harmonic oscillator's potential surrounding?

For infinite wall,ψ(0)=0 ,ψ(L)=0, L is the width of the well. Then what is the meaning when we says ψ(0)=0, is that mean there is no particle at x=0? If the wavefunction square doesn't give the probability of anything, then why is ψ(0)=0 correct?

4. Oct 19, 2013

### vanhees71

If a particle is prepared in a state described by the wave function $\psi(x)$ (which implies that we work in non-relativstic quantum theory!), then $|\psi(x)|^2$ is the probabilitity distribution to find a particle at position $x$, i.e., the probability to find the particle in an infinitesimal interval of length $\mathrm{d} x$ around position x is $\mathrm{d} x |\psi(x)|^2$.

5. Oct 19, 2013

### Simon Bridge

We imagine the particle experiences a force from somewhere.
The potential describes how that force changes with the position of the particle without saying anything about what causes the force.

It means the amplitude of the wave-function is zero there. The wavefunction is totally abstract.

Because that way it will be a solution to the Schrodinger Equation. Because that is how wave-functions behave if they are to lead to physically meaningful results. It's still not a probability function.

The wavefunction itself does not give probabilities, but you can use it to construct the probability density function for position (see vanhees above). Do you know how probability density functions work?

The probability of finding the particle at a particular position x is zero. The classical idea that a particle may occupy a particular position at a particular time is one of those ideas you have to give up.

You can only measure a position to be within some range of values.

The probability that the particle will be found between x=a and x=b : a<b is: $$P(a<x<b)=\int_a^b\psi^\star (x)\psi (x)\; \mathrm{d}x$$

Are you teaching yourself quantum mechanics or are you doing a course?

Last edited: Oct 19, 2013
6. Oct 19, 2013

### Outrageous

Can you please give me one application when do we apply harmonic oscillator?

Because this is correct when we get the eigenvalue from experiment. This is what I read say. Correct?

A continuous graph that is ψ(x) square against x which tells us the probability of finding a particle within some range of values.

Thanks vanhees and Simon Bridge

I am taking the quantum mechanics course.

7. Oct 19, 2013

### Outrageous

I know equation 1 is used to solve the schrodinger equation 2.

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8. Oct 19, 2013

### Outrageous

But for harmonic oscillator,what are the equations 3 ? E= [ψ+(a-)][ψ-(a+)]?
Then what leads to equations 4 ?
Reading from introduction to quantum mechanics - D.Griffiths.

9. Oct 19, 2013

### Outrageous

For the previous post

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10. Oct 19, 2013

### Simon Bridge

molegules in a gas, atoms in a lattice, nucleons in the low-energy regime can all be modelled using HO potentials.

You get a measurement from an experiment.
Whether that is an eigenvalue of the system depends on the system and the measurement.

Can be used to find the probability of finding the particle within a range of positions.

... the concept of a wavefunction should have been explained to you as part of the course i]before[/i] you have to work out potential wells.
You usually need to know something about classical probability density functions before you start too.
I think you need to revise your earlier coursework.
It can take several goes to "get" it though ... QM is very non-intuitive at first.

11. Oct 19, 2013

### Simon Bridge

Eq1 is the set of solutions to the Shrödinger equation, for the infinite square well.
The stuff between eq2 and eq1 is what you used to solve it ... along with the boundary conditions and the potential energy function.

I don't have that text.

Solving for the HO is more complicated than for the ISW.
http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

Looking at your attachments - your notes seem to use the ladder operator method. (see link above)
You should already have been introduced to the concept of an operator and shown what the ladder operators do.

You may understand it better using the spectral approach.