# Simple Harmonic Oscillator Squeezing

• t0pquark
In summary, the conversation discusses the process of obtaining Equation 5.27 from Equation 5.26 in the lecture notes for MIT's course on quantum physics. The speaker is struggling with the substitution of creation and annihilation operators to rearrange the equations, but is unable to obtain the expected result. It is pointed out that Equation 5.27 is a repetition of Equation 5.12 and does not follow from 5.26.
t0pquark
Homework Statement
How to derive ##(\hat{a}cosh\gamma + \hat{a^\dagger}sinh\gamma \vert 0_{\gamma} \rangle = 0 ## from ##\vert 0_{\gamma} \rangle \equiv \frac{1}{\sqrt{cosh\gamma}} exp(-\frac{1}{2}tanh\gamma \hat{a^\dagger}\hat{a^\dagger} \vert 0 \rangle##
Relevant Equations
Creation operator: ##\hat{a}_2 = \hat{a}_1cosh\gamma - \hat{a}_1^\dagger sinh\gamma##
Annihilation operator: ##\hat{a}_2^\dagger = -\hat{a}_1sinh\gamma + \hat{a}_1^\dagger cosh\gamma##
Where ##e^\gamma \equiv \sqrt{\frac{m_1 \omega_1}{m_2 \omega_2}}##
I'm working through https://ocw.mit.edu/courses/physics...all-2013/lecture-notes/MIT8_05F13_Chap_06.pdf, and I'm stumped how they got from Equation 5.26 (##\vert 0_{\gamma} \rangle \equiv \frac{1}{\sqrt{cosh\gamma}} exp(-\frac{1}{2}tanh\gamma \hat{a^\dagger}\hat{a^\dagger} \vert 0 \rangle##) to Equation 5.27 (##(\hat{a}cosh\gamma + \hat{a^\dagger}sinh\gamma) \vert 0_{\gamma} \rangle = 0 ##).
I've tried substituting in the creation (##\hat{a}_2 = \hat{a}_1cosh\gamma - \hat{a}_1^\dagger sinh\gamma##) and annihilation ((##\hat{a}_2^\dagger = -\hat{a}_1 sinh\gamma + \hat{a}_1^\dagger cosh\gamma##) operators and then rearranging, but I can't get what they have.

Equation 5.27 does not follow from 5.26. It is a repetition of 5.12.

Abhishek11235

## 1. What is a simple harmonic oscillator?

A simple harmonic oscillator is a type of mechanical system that follows a specific pattern of motion where the force acting on the system is directly proportional to the displacement from its equilibrium position. This results in a repetitive back-and-forth motion around the equilibrium point.

## 2. What is squeezing in the context of a simple harmonic oscillator?

Squeezing in a simple harmonic oscillator refers to the phenomenon where the uncertainty in the position of the oscillator decreases while the uncertainty in its momentum increases, or vice versa. This is achieved by manipulating the initial conditions or parameters of the oscillator.

## 3. How is squeezing beneficial in a simple harmonic oscillator?

Squeezing can be beneficial in a simple harmonic oscillator because it can lead to a more precise measurement of the oscillator's position or momentum. It can also help to reduce the effects of external disturbances on the oscillator's motion.

## 4. Can the squeezing effect be reversed in a simple harmonic oscillator?

Yes, the squeezing effect in a simple harmonic oscillator can be reversed by changing the initial conditions or parameters of the oscillator back to their original values. This would result in an increase in uncertainty in one variable while decreasing the uncertainty in the other variable.

## 5. How is squeezing related to Heisenberg's uncertainty principle?

Squeezing in a simple harmonic oscillator is related to Heisenberg's uncertainty principle, which states that it is impossible to simultaneously know the exact position and momentum of a particle. By squeezing one variable, the uncertainty in the other variable must increase, thus demonstrating the principle.

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