Solving the Shroedinger equation for a harmonic oscillator potential

In summary, Griffith solves the Schrodinger equation for a harmonic oscillator potential using the power series method. First he rewrites the shroedinger equation in the form d^2ψ/dε^2 = (ε^2 - K)ψ , where ε= x√(mw/hbar) and K=2E/(ω hbar ) . Then he says that at large ε, we can approximate the equation to be d^2ψ/dε^2 ≈ ε^2ψ . So ψ≈Ae^(-ε^2 /2 ) + Be^(ε^2 /2 ) . But at large ε
  • #1
user3
59
0
Hello,

I have been studying Introduction to Quantum Mechanics by Griffith and in a section he solves the Schrodinger equation for a harmonic oscillator potential using the power series method. First he rewrites the shroedinger equation in the form d^2ψ/dε^2 = (ε^2 - K)ψ , where ε= x√(mw/hbar) and K=2E/(ω hbar ) .
Then he says that at large ε, we can approximate the equation to be d^2ψ/dε^2 ≈ ε^2ψ . So ψ≈Ae^(-ε^2 /2 ) + Be^(ε^2 /2 ) . But at large ε, we have to remove the Be^(ε^2 /2 ) term because otherwise, the wave function wouldn't be normalizable: ψ≈Ae^(-ε^2 /2 )

and then he does something that I don't understand :

" ψ(ε) → ( ) e^(-ε^2 /2 ) at large ε

This suggests we "peel off" the exponential part,

ψ(ε) = h(ε)e^(-ε^2 /2 )

"

and then he goes on to solve the h(ε) rather than the ψ(ε) : h(ε) = ∑aj ε^j


Why did he do that? Why not from the very beginning assume that ψ(ε)= ∑aj ε^j and find a recursion formula for that ?
 
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  • #2
Because he wanted to motivate the decision to use a power series. In my copy Griffith's even makes a footnote stating that this is, as you reasoned, the idea behind solving diff. eq's using power series.
 
  • #3
but I tried rewriting the Shroedinger equation as d^2ψ/dε^2 - (ε^2 - K)ψ = 0 and then assumed ψ=∑aj ε^j , then substituted in the Shroedinger equation but got a different recursion formula :
a_(j+2) = (a_(j-2) - a_(j)K ) / (j+2)(j+1) . How can I deduce that K must equal 2j+1 from this recursion relation ?
 
  • #4
user3 said:
but I tried rewriting the Shroedinger equation as d^2ψ/dε^2 - (ε^2 - K)ψ = 0 and then assumed ψ=∑aj ε^j , then substituted in the Shroedinger equation but got a different recursion formula :
a_(j+2) = (a_(j-2) - a_(j)K ) / (j+2)(j+1) . How can I deduce that K must equal 2j+1 from this recursion relation ?
You can't. That's because it is a three-term recursion relation (i.e. it involves a's with three different subscripts).

Which is exactly the purpose of factoring out the exponential - to lead to a differential equation that can be solved by a two-term recursion relation and therefore has polynomial solutions.
 
  • #5
I did not separate out the Gaussian potential once about 5 or 6 years ago. Like Bill_K, I got a three term recursion relation. Best to do as Griffith or other QM texts recommend.
 
  • #6
Well the reason he does that is that he needs the exponential to save the normalizability of the function for ε going to infinity...
Then what else could someone think as the general solution that has exponential damping in the upper limit?
In general it's the sum of powers of ε, since their growing rate is canceled out by the exponential's decrease.

Almost the same approaches you can find in Hydrogen atom solutions...
 

FAQ: Solving the Shroedinger equation for a harmonic oscillator potential

1. What is the Schrodinger equation?

The Schrodinger equation is a mathematical equation that describes how the quantum state of a physical system changes with time. It is a fundamental equation of quantum mechanics and is used to calculate the probability of finding a particle in a particular state.

2. What is a harmonic oscillator potential?

A harmonic oscillator potential is a type of potential energy function that describes the behavior of a particle that is subject to a restoring force proportional to its displacement from an equilibrium position. It is commonly used to model the behavior of atoms, molecules, and other systems in physics.

3. How do you solve the Schrodinger equation for a harmonic oscillator potential?

The Schrodinger equation for a harmonic oscillator potential can be solved using various mathematical techniques, such as the method of undetermined coefficients or the ladder operator method. These methods involve solving for the wave function, which describes the quantum state of the system, and determining the energy levels of the system.

4. What are the applications of solving the Schrodinger equation for a harmonic oscillator potential?

The solutions to the Schrodinger equation for a harmonic oscillator potential have many applications in physics, including understanding the behavior of atoms and molecules, modeling the motion of particles in a potential well, and predicting the energy levels of quantum systems.

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

While the Schrodinger equation is a powerful tool for understanding quantum systems, it does have some limitations. For example, it assumes that the potential energy is known and constant, which may not always be the case in real-world systems. Additionally, the solutions to the equation may not always accurately predict the behavior of complex systems with multiple particles.

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