Forced Harmonic Oscillator with Path Integral

In summary, the transition amplitude of the forced harmonic oscillator can be found by taking the Fourier transformation of the path, plugging the path into the functional, separating out a path-independent term and a path-dependent term, and setting the whole expression equal to 1 when the source is zero.
  • #1
Phileas.Fogg
32
0
Hello,
how do I compute the transition amplitude of the forced harmonic oscillator with the method of path integration?

Regards,
Mr. Fogg
 
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  • #2
Do you mean the vacuum to vacuum transition amplitude? You take the Fourier transformation of the path, plug the path into the functional, complete the square to separate out a path-independent term and a path-dependent term, shift the path so that the path-dependent term is no longer a function of the source, then set the whole expression equal to 1 when the source is zero to correctly normalize it, and this will show you that the transition amplitude is the path-independent term.
 
  • #3
Phileas.Fogg said:
Hello,
how do I compute the transition amplitude of the forced harmonic oscillator with the method of path integration?

Regards,
Mr. Fogg

Make the ansatz: [tex]x=\bar{x}+y[/tex], where [tex]\bar{x}[/tex] is the classical solution. Then we can write the "variation" y as a Fourier expansion in time. Thus the kernel K[b,a] is

[tex]K[b,a]=\e^{\frac{i}{\hbar}S_{cl}[b,a]}F[T][/tex]

and normalization constant in F(T) could be found by the limit w->0 (free particle case). The action of the classical trajectory could be obtained using the Green function. We have the system: x''+w^2x=f(t)/m and BC x(ta)=Xa and x(tb)=Xb. The Lagrangian could then be obtained if we know x(t) which is given as:

[tex]\bar{x}(t)=\int_{ta}^{tb}G(t,\chi)f(t)/m\cdot d\chi[/tex]

Hope this helps you a bit...
Per
 
  • #4
HI!
Thanks for your help.

The Lagrangian is
[tex] L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} m \omega^2 x^2 + J(t) x [/tex]

I started with:

[tex]x = x_{cl} + y[/tex]

and showed that the transition amplitude can be written as

[tex] U(x_a,x_b,t_b) = \exp(-\frac{i}{\hbar}S(x_{cl})) + \int_{y(0)=0}^{y(t_{b})=0} [dy]\exp(-\frac{i}{\hbar} \int_0^{t_{b}} dt \frac{1}{2}m (\dot{y}^2 - \omega^2 y^2)[/tex]

Now I wan't to show with Path Integration that the transition amplitude is then

[tex] U(x_a, x_b, t_b) = \lim\limits_{N \rightarrow \infty}\frac{m}{2 \pi i \hbar \epsilon Q_{N-1}} e^{\frac{i}{\hbar} S(x_{cl})} \quad \text{(1.1)}[/tex]

with [tex] \epsilon = \frac{t_b}{N} \quad Q_{N-1} = det(A)[/tex]

A is a quadratic Matrix.

I don't know, how to get the Matrix-Expression (1.1) of the transition amplitude.

Regards,
Mr.Fogg
 
Last edited:
  • #5
Maybe this link will help:

http://en.wikipedia.org/wiki/Gaussian_integral#n-dimensional_and_functional_generalization

The det(A) doesn't really matter.

As I think already has been mentioned, for your particular problem (no more than quadratic in q), you can also find the classical solution to your Lagrangian, and plug that solution into your Lagrangian. The classical solution should be:

[tex] q(t)=\int \frac{d^4n}{(2\pi)^4} \frac{f(n)}{m(\omega^2-n^2)}e^{-int}[/tex]

where f(n) is the Fourier-Transform of f(t).
 

Related to Forced Harmonic Oscillator with Path Integral

1. What is a forced harmonic oscillator?

A forced harmonic oscillator is a physical system that experiences a restoring force that is directly proportional to the displacement from its equilibrium position, and is also subject to an external force or driving force.

2. How is the path integral used in studying forced harmonic oscillators?

The path integral is a mathematical tool used to calculate the probability amplitude for a quantum mechanical system to transition from one state to another. In the case of a forced harmonic oscillator, the path integral can be used to calculate the probability of the system's trajectory or path in phase space.

3. What is the difference between a classical and quantum forced harmonic oscillator?

In a classical forced harmonic oscillator, the system's energy is continuous and can take on any value, while in a quantum forced harmonic oscillator, the energy is quantized and can only take on certain discrete values. Additionally, the path integral is used to study the quantum system, while classical mechanics equations are used to study the classical system.

4. What is the significance of the path integral in studying forced harmonic oscillators?

The path integral provides a powerful method for calculating the probability amplitudes and expected values of observables in quantum systems, including forced harmonic oscillators. It allows for a more intuitive understanding of the system's behavior and can be used to study both the classical and quantum aspects of the system.

5. How does the path integral approach differ from other methods of studying forced harmonic oscillators?

The path integral approach is a non-perturbative method, meaning it can be used for systems with strong interactions and does not require making small approximations. Other methods, such as perturbation theory, may only be applicable for weakly interacting systems. The path integral also allows for a more visual representation of the system's behavior in phase space, making it a valuable tool for understanding complex systems like forced harmonic oscillators.

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