Perturbed in the harmonic oscillator

In summary, the conversation revolved around the topic of writing a power series form for part a of a given problem. The expert suggests expanding (1 + \epsilon)^{1/2} in a binomial series and mentions that the textbook may define H', E_n^1, and \psi_n^0 to be used in the Schrodinger equation. They also mention the importance of taking an inner product with \langle\psi_n^0| and the self-adjoint property of H_0. The conversation then shifts to discussing the incorrectness of H' and the possibility of a mistake in the evaluation of the integral.
  • #1
Fatimah od
22
0

Homework Statement



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Homework Equations



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The Attempt at a Solution



for part a I do not know how to write it in power series form ?

for part b :
I chose the perturbed H' is v(x)= (1+ε )K x^2 /2
then I started integrate E_1 = ∫ H' ψ^2 dx

the problem was , the result equals to ∞ !

shall I integrate in the interval [ 0,∞ ] !
 
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  • #2
Fatimah od said:
for part a I do not know how to write it in power series form ?

You are given [itex]E_n = (n + \frac12)\hbar\sqrt{k/m}[/itex]. [itex]k[/itex] is now [itex]k(1+\epsilon)[/itex], so [itex]E_n[/itex] is now
[tex]
E_n = \left(n + \frac12\right)\hbar\sqrt{\frac km(1 + \epsilon)} =
\left(n + \frac12\right)\hbar\sqrt{\frac km}(1 + \epsilon)^{1/2}
[/tex]
Since [itex]|\epsilon| < 1[/itex], [itex](1 + \epsilon)^{1/2}[/itex] can be expanded in a binomial series.
 
  • #3
ok , I do it thanks

any help for second part please
 
  • #4
Fatimah od said:
ok , I do it thanks

any help for second part please

I can't help you with that unless you tell me how your textbook defines [itex]H'[/itex], [itex]E_n^1[/itex] and [itex]\psi_n^0[/itex]. I am prepared to guess that you have
[tex]E_n = E_n^0 + \epsilon E_n^1 + O(\epsilon^2)\\
\psi_n = \psi_n^0 +\epsilon\psi_n^1 + O(\epsilon^2)[/tex]
but I'm not prepared to guess what [itex]H'[/itex] might be. However I suspect the method is to substitute the above into the Schrodinger equation to get
[tex]
(E_n^0 + \epsilon E_n^1)(\psi_n^0 + \epsilon \psi_n^1) = -\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} (\psi_n^0 + \epsilon\psi_n^1) + \frac{kx^2}{2}(1 + \epsilon)(\psi_n^0 +\epsilon\psi_n^1)
[/tex]
and then require that the coefficients of [itex]\epsilon^0[/itex] and [itex]\epsilon^1[/itex] should vanish. At some stage you may want to take an inner product with [itex]\langle\psi_n^0|[/itex], and recall that
[tex]H_0 = -\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \frac{kx^2}{2}[/tex]
is self-adjoint ([itex]\langle f | H_0 | g \rangle = \langle g | H_0 | f \rangle[/itex] for all [itex]f[/itex] and [itex]g[/itex]).
 
  • #5
Your H' is wrong, or you're not expressing yourself clearly. H' is the difference between the complete Hamiltonian and the unperturbed Hamiltonian.

Show us how you're getting that the integral diverges. The problem seems to lie in your evaluation of the integral.
 

1. What is a perturbed harmonic oscillator?

A perturbed harmonic oscillator is a physical system that follows the same equations of motion as a simple harmonic oscillator, but with an added external force or disturbance that causes deviations from the ideal simple harmonic motion.

2. How does perturbation affect the behavior of a harmonic oscillator?

Perturbation can cause variations in the frequency, amplitude, and phase of a harmonic oscillator. These variations depend on the strength and nature of the perturbation, as well as the initial conditions of the oscillator.

3. What are some examples of perturbed harmonic oscillators?

Examples of perturbed harmonic oscillators include a pendulum with air resistance, an electrical circuit with resistance, and a mass-spring system with friction.

4. How is the perturbed harmonic oscillator described mathematically?

The perturbed harmonic oscillator can be described mathematically using the second-order differential equation: m(d^2x/dt^2) + kx + F(x,t) = 0 where m is the mass, k is the spring constant, x is the displacement from equilibrium, t is time, and F(x,t) is the external force or disturbance.

5. How can we analyze the behavior of a perturbed harmonic oscillator?

There are several methods for analyzing the behavior of a perturbed harmonic oscillator, including numerical simulations, approximations such as the perturbation method or small-angle approximation, and graphical techniques such as phase portraits and energy diagrams.

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