Solving Schrödinger Eq. for Harmonic Oscillator

In summary, the conversation discusses how to arrive at equation 2.56 by performing substitutions. The first substitution is for x^2 and the second substitution is for the second derivative in x. The conversation also addresses the question of why the second derivative in xi does not vanish, as well as how to continue the derivation using the product rule.
  • #1
Azorspace
9
0

Homework Statement



I wonder if someone could help me to arrive at equation 2.56 by performing the substitutions. Please see the attachment

Homework Equations



Please see the attachment for this part. and also for the attempt of a solution.
 

Attachments

  • Harmonic oscillator.doc
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  • #2
You don't show your work, so I cannot figure out what you did. This is what I would do. There are two substitutions in the original Schrodinger equation. First replace

[tex]x^{2}=\frac{\hbar}{m \omega}\xi^{2}[/tex]

Secondly, you need to replace the second derivative in x with the second derivative in ξ. Use the chain rule

[tex]\frac{d}{dx}=\frac{d \xi}{dx}\frac{d}{d \xi}=\sqrt{\frac{m \omega}{\hbar}}\frac{d}{d \xi}[/tex]

Repeat to get the second derivative and plug in. It should come out as advertised.
 
  • #3
First, thanks for replying. Well the main question here is why the next happens:

[tex]\frac{1}{mw}\frac{d^2\psi}{dx^2}=\frac{d^2\psi}{d\xi^2}[\tex]

Because if you take the second derivative of xi, you get this:

[tex]\frac{d\xi}{dx}=\sqrt{\frac{m \omega}{\hbar}}[\tex]

[tex]\Rightarrow\frac{d^2\xi}{dx^2}=0[\tex]

Because:

[tex]\sqrt{\frac{m \omega}{\hbar}}=constant[\tex]

so you can't get there by taking the second derivative of xi or x.

Sorry i don't know how to make tex to work but i have attached the reply.
 

Attachments

  • reply.pdf
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Last edited:
  • #4
I cannot see the pdf yet (pending approval) but the LateX code shows that you think that because

[tex]
\frac{d}{dx}=\sqrt{\frac{m \omega}{\hbar}}\frac{d}{d \xi}[/tex]

you think that the second derivative vanishes because

[tex]
\sqrt{\frac{m \omega}{\hbar}}= constant[/tex]

That is not true. Remember that the derivative operates on a function. Let's put one in (I suggest that you do so until you get used to the algebra of operators) and see what happens. From the top

[tex]\frac{d \psi}{dx}=\sqrt{\frac{m \omega}{\hbar}}\frac{d \psi}{d \xi}[/tex]

Then taking the derivative with respect to x once more,

[tex]\frac{d^{2} \psi}{dx^{2}}=\sqrt{\frac{m \omega}{\hbar}}\frac{d}{dx}(\frac{d \psi}{d \xi})=\sqrt{\frac{m \omega}{\hbar}}\sqrt{\frac{m \omega}{\hbar}}\frac{d}{d \xi}(\frac{d \psi}{d \xi})[/tex]

Note that when you take the derivative once more, a product rule is implied where only one term in the product is constant. Can you finish it now?
 
  • #5
Yes, i got it now, thanks for the help
 

1. What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the evolution of a quantum system over time. It was developed by Austrian physicist Erwin Schrödinger in 1926.

2. What is a harmonic oscillator?

A harmonic oscillator is a physical system that exhibits a characteristic pattern of motion, where the restoring force is directly proportional to the displacement from equilibrium. Examples of harmonic oscillators include a mass attached to a spring or a pendulum.

3. How is the Schrödinger equation used to solve for a harmonic oscillator?

The Schrödinger equation is used to calculate the wave function of a quantum system, which describes the probability of finding the system in a particular state. For a harmonic oscillator, the Schrödinger equation can be solved using mathematical techniques such as the method of separation of variables or the ladder operator method.

4. Why is solving the Schrödinger equation for a harmonic oscillator important?

Solving the Schrödinger equation for a harmonic oscillator allows us to understand and predict the behavior of quantum systems with harmonic potential. This is important in various fields such as chemistry, material science, and quantum computing.

5. Are there any limitations to solving the Schrödinger equation for a harmonic oscillator?

Yes, there are limitations to solving the Schrödinger equation for a harmonic oscillator. The equation assumes that the harmonic potential is the only force acting on the system, which may not always be the case in real-world systems. Additionally, the solutions to the Schrödinger equation may not always accurately predict the behavior of systems with strong interactions or at very high energies.

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