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

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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 ?

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  • #2
Simon Bridge
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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.
 
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It means that the potential function has the same maths as that for an ideal harmonic oscillator.


The wavefunction does not give the probability of anything.
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?
 
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vanhees71
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If a particle is prepared in a state described by the wave function [itex]\psi(x)[/itex] (which implies that we work in non-relativstic quantum theory!), then [itex]|\psi(x)|^2[/itex] is the probabilitity distribution to find a particle at position [itex]x[/itex], i.e., the probability to find the particle in an infinitesimal interval of length [itex]\mathrm{d} x[/itex] around position x is [itex]\mathrm{d} x |\psi(x)|^2[/itex].
 
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  • #5
Simon Bridge
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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?
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.

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

If the wavefunction square doesn't give the probability of anything, then why is ψ(0)=0 correct?
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?
 
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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.
Can you please give me one application when do we apply harmonic oscillator?

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

Do you know how probability density functions work?
A continuous graph that is ψ(x) square against x which tells us the probability of finding a particle within some range of values.



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

Are you teaching yourself quantum mechanics or are you doing a course?
I am taking the quantum mechanics course.
 
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I know equation 1 is used to solve the schrodinger equation 2.
 

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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.
 
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For the previous post
 

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Simon Bridge
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Can you please give me one application when do we apply harmonic oscillator?
molegules in a gas, atoms in a lattice, nucleons in the low-energy regime can all be modelled using HO potentials.

Because this is correct when we get the eigenvalue from experiment. This is what I read say. Correct?
You get a measurement from an experiment.
Whether that is an eigenvalue of the system depends on the system and the measurement.

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

I am taking the quantum mechanics course.
... 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.
 
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Simon Bridge
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I know equation 1 is used to solve the schrodinger equation 2.
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.

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.
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.
(also linked from the link above)
 
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Thanks
 

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