Harmonic oscilator Tamvakis

In summary, the conversation discusses how to solve the problem of finding the commutator between the Hamiltonian operator and the position operator squared. The resulting commutator is shown to be equal to a combination of the Hamiltonian operator and the position operator, with factors of Planck's constant and the mass and angular frequency of the system. Several attempts at solving this problem are discussed, with the final solution involving expanding the commutator and using the commutation relation between position and momentum.
  • #1
LagrangeEuler
717
20

Homework Statement


## H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2##
Show that
##[H,[H,x^2]]=(2\hbar\omega)^2x^2-\frac{4\hbar^2}{m}H##


Homework Equations


##[x,p]=i\hbar##


The Attempt at a Solution


I get
##[H,x^2]=-\frac{i\hbar}{m}(px+xp)##
what is easiest way to solve this problem?
 
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  • #2
Show your work.
 
  • #3
##[H,x^2]=[\frac{p^2}{2m},x^2]=\frac{1}{2m}[p^2,x^2]=p[p,x^2]+[p,x^2]p##
##[p,x^2]=-2i\hbar x##
from that
##[H,x^2]=-2i\hbar px-2i\hbar xp##
from that
##[H,[H,x^2]]=p[\frac{p^2}{2m},-2i\hbar x]+[\frac{p^2}{2m},-2i\hbar x]p+p[\frac{1}{2}m\omega^2x^2,-2i\hbar x]+[\frac{1}{2}m\omega^2x^2,-2i\hbar p]x##
from that
##[H,[H,x^2]]=-4p^2\frac{\hbar^2}{m}-4m\omega^2i\hbar x##
you don't get this result.
 
Last edited:
  • #4
First part is same. But its to hard. I tried
[tex][H,[H,x^2]]=[H,Hx^2-x^2H]=H[H,x^2]-[H,x^2]H=H(-2i\hbar)x+2i\hbar xH[/tex]
 
  • #5
I solve it.
 
Last edited:

1. What is a harmonic oscillator Tamvakis?

A harmonic oscillator Tamvakis is a specific type of oscillating system that follows the mathematical model of a harmonic oscillator. It was developed by Greek physicist George Tamvakis, and is commonly used in physics and engineering to analyze the behavior of oscillating systems.

2. How does a harmonic oscillator Tamvakis work?

A harmonic oscillator Tamvakis works by following the principles of a harmonic oscillator, which is a system that experiences a restoring force proportional to its displacement from equilibrium. This force causes the system to oscillate back and forth around its equilibrium point.

3. What are some real-life examples of harmonic oscillator Tamvakis?

Some common examples of systems that can be modeled using a harmonic oscillator Tamvakis include a mass-spring system, a pendulum, and a vibrating guitar string. These systems all exhibit harmonic motion, meaning they follow the same mathematical model as a harmonic oscillator Tamvakis.

4. How is a harmonic oscillator Tamvakis different from a regular harmonic oscillator?

A harmonic oscillator Tamvakis differs from a regular harmonic oscillator in that it takes into account the effects of damping, which is the gradual loss of energy in an oscillating system. This makes it a more realistic model for many real-life systems, since most oscillating systems experience some form of damping.

5. What are the applications of a harmonic oscillator Tamvakis?

A harmonic oscillator Tamvakis has many practical applications, including analyzing the behavior of mechanical systems like springs and pendulums, predicting the motion of molecules in quantum mechanics, and understanding the behavior of electrical circuits in electronics. It is also used in many fields of engineering to design and optimize systems that exhibit harmonic motion.

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