Simple Harmonic Oscillator (time independant Schrodingers)

In summary, the given problem involves a particle with mass m confined by a one-dimensional harmonic oscillator potential V(x)=Cx2. By substituting this potential into the time-independent Schrodinger equation, it is shown that the spatial wavefunction \psi(x)=Axe-ax2 can be a possible solution if the constant a has a certain value. To find this value in terms of C, m, and hbar, the TISE is solved using the given wavefunction. The chosen value of a should result in the TISE being satisfied for all values of x and should also remove any x-dependence from the equation, thus giving the energy eigenvalue for the given state. After solving for a, the final energy eigenvalue is
  • #1
indie452
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0

Homework Statement



Particle mass m is confined by a one dimensional simple harmonic oscillator potential V(x)=Cx2, where x is the displaecment from equilibrium and C is a constant

By substitution into time-independant schrodingers with the potential show that
[tex]\psi[/tex](x)=Axe-ax2
is a possible spatial wavefunction for this particle provided constant a has a certain value.
Find a in term os C, m, h bar (\h)
What the corresponding energy eigen value

The Attempt at a Solution



[tex]\psi[/tex](x)= Axe-ax2
d/dx [tex]\psi[/tex](x)= [A - 2Aax2]*e-ax2
d2/d2x [tex]\psi[/tex](x)= [4a2Ax3 - 6aAx]*e-ax2

so shrodingers:


[tex]\frac{-hbar^2}{2m}[/tex]*[4a2x2 - 6a] + Cx2 = E

Ive gotten to this bit but i don't understand what to do, i have read the 4 page solution in Eisburg&Resnick and spoke to my lecturer, but my lecturer has said that i don't need to do the long winded complete solution and that i is also not necessary to evaluate the normalisation constant A.
I just don't know how to show that the psi above is a possible wavefunction

help or advice on how to do this problem would be much appreciated

ps I have tried looking at and doing some other example but they all had the potential as V(x)= 1/2 Cx2, this seemed to be the standard example. I also tried using the substitution w2= (c/m) and this would give a potential V(x)= mw2x2, but i didnt know if this was right as the question ask for a in terms of C (and m and hbar)
 
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  • #2
indie452 said:
so shrodingers:

[tex]\frac{-hbar^2}{2m}[/tex]*[4a2x2 - 6a] + Cx2 = E

so i think you're pretty much there, so assuming you have subbed into the TISE correctly (which looks ok to me...), then how can you "choose" a, so that the above equation i satisfied?
 
  • #3
well the thing is i don't know when it would be satisfied other than the LHS=RHS but i don't know how i would get E on the LHS, unless i made the sub that E=hbar*w
 
  • #4
I don't see a w anywhere... but there should be one value of a to satisfy TISE for all x, and what if the TISE is telling you the energy of the given state...
 
  • #5
As lanedance said, you're nearly there. You do know that E is a constant (but don't know it yet), i.e. it is independent of x.
 
  • #6
thats what I am having trouble with - knowing when it is satisfied

is the LHS supposed to equal the RHS?
i tried a=[tex]\sqrt{\frac{mc}{hbar}}[/tex]

and i got

Cx2(1-2hbar) + [tex]\frac{3x(hbar)\sqrt{hbar*mC}}{m}[/tex] = E

but i don't know what to do with it...is my choice for a even right?
How do you choose the value, is it a guess? or is there some method?
 
  • #7
no, i don't think so, the Energy of a eigenstate in the TISE, should be a constant scalar value (independent of x), this should help you choose a to remove any x dependence
 
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  • #8
ok i tried this

[tex]\frac{(hbar)^2}{2m}[/tex]*[6a - 4ax2 + [tex]\frac{2m}{(hbar)^2}[/tex]Cx2] = E

6a + x2*[[tex]\frac{2m}{(hbar)^2}[/tex]Cx2 - 4a] = [tex]\frac{2m}{(hbar^2}[/tex]Cx^2]E

i tried a= mC/2(hbar)2

this gave me

[tex]\frac{6mC}{2(hbar)^2}[/tex] + x2[[tex]\frac{2mC}{(hbar)^2}[/tex] - [tex]\frac{4mC}{2(hbar)^2}[/tex]] = [tex]\frac{2m}{(hbar)^2}[/tex]Cx^2]E

=[tex]\frac{6mC}{2(hbar)^2}[/tex] = [tex]\frac{2m}{(hbar)^2}[/tex]Cx^2]E

E=[tex]\frac{3C}{2}[/tex]
 
  • #9
i can't follow your equations i the last post... but i think you've got the idea
 
  • #10
also do you know you can write longer equations in tex? eg.

[tex] \frac{\hbar^2}{2m}(4a^2x^2 - 6a) + Cx^2 = E [/tex]
 
  • #11
sorry made a mistake there, this is what i meant

[tex]\frac{(hbar)^2}{2m}[/tex]*[6a - 4ax2 + [tex]\frac{2m}{(hbar)^2}[/tex]Cx2] = E

6a + x2*[[tex]\frac{2m}{(hbar)^2}[/tex]Cx2 - 4a] = [tex]\frac{2m}{(hbar)^2}[/tex]E

i tried a= mC/2(hbar)2

this gave me

[tex]\frac{6mC}{2(hbar)^2}[/tex] + x2[[tex]\frac{2mC}{(hbar)^2}[/tex] - [tex]\frac{4mC}{2(hbar)^2}[/tex]] = [tex]\frac{2m}{(hbar)^2}[/tex]E

=[tex]\frac{6mC}{2(hbar)^2}[/tex] = [tex]\frac{2m}{(hbar)^2}[/tex]E

E=[tex]\frac{3C}{2}[/tex]
 

FAQ: Simple Harmonic Oscillator (time independant Schrodingers)

What is a simple harmonic oscillator?

A simple harmonic oscillator is a system that exhibits periodic motion around an equilibrium point. It can be described mathematically using the time independent Schrodinger equation, which is a fundamental equation in quantum mechanics.

What is the significance of the time independent Schrodinger equation in the study of simple harmonic oscillators?

The time independent Schrodinger equation provides a way to calculate the energy levels and corresponding wavefunctions of a simple harmonic oscillator. This allows for a deeper understanding of the system's behavior and can be used to make predictions about its future motion.

How is the time independent Schrodinger equation derived for a simple harmonic oscillator?

The time independent Schrodinger equation is derived by applying the principles of quantum mechanics to the classical harmonic oscillator system. This involves using the Hamiltonian operator to represent the total energy of the system and solving for the eigenvalues and eigenvectors of the operator.

Can the time independent Schrodinger equation be used for any type of oscillator?

Yes, the time independent Schrodinger equation can be used to describe the behavior of any type of oscillator, as long as the potential energy function is known. This includes not only simple harmonic oscillators, but also more complex systems such as anharmonic oscillators.

What are the limitations of using the time independent Schrodinger equation to study simple harmonic oscillators?

The time independent Schrodinger equation assumes that the system is in a stationary state, meaning that the energy levels and wavefunctions do not change with time. This is not always the case for real systems, which may experience damping or external forces that cause the energy levels to shift. Additionally, the equation only applies to non-relativistic systems and does not take into account any relativistic effects.

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