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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

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- Thread starter Outrageous
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- #1

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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

Simon Bridge

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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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- #5

Simon Bridge

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We imagine the particle experiences a force from somewhere.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?

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.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?

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.If the wavefunction square doesn't give the probability of anything, then why is ψ(0)=0 correct?

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

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

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

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

Thanks vanhees and Simon BridgeThe 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: $$P(a<x<b)=\int_a^b\psi^\star (x)\psi (x)\; \mathrm{d}x$$

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

I am taking the quantum mechanics course.

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Then what leads to equations 4 ?

Reading from introduction to quantum mechanics - D.Griffiths.

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- #10

Simon Bridge

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

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

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.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 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.I am taking the quantum mechanics course.

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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Eq1 is the set of solutions to the Shrödinger equation, for the infinite square well.I know equation 1 is used to solve the schrodinger equation 2.

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.

Then what leads to equations 4 ?

Reading from introduction to quantum mechanics - D.Griffiths.

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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