# Quantum Harmonic Oscillator - What is the Temperature of the system?

• obi-wan
Thanks again for your help!In summary, a quantum mechanical harmonic oscillator with resonance frequency ω and mean excitation energy of 0.3ħω in an environment at temperature T can be determined using the partition function and the equation E= -δ/δβ ln[Z(β)]. The temperature of the system can be found by solving for β, which can then be used to determine the temperature in units of the Einstein-temperature ΘE = ħω/kB. The final temperature is approximately 0.6208ΘE.
obi-wan

## Homework Statement

A quantum mechanical harmonic oscillator with resonance frequency ω is placed in an environment at temperature T. Its mean excitation energy (above the ground state energy) is 0.3ħω.
Determine the temperature of this system in units of its Einstein-temperature ΘE = ħω/kB

## Homework Equations

Partition-Function= E= -δ/δβ ln[Z(β)]

## The Attempt at a Solution

I have used the Partiton-Function, E= -δ/δβ ln[Z(β)] = -δ/δβ ln[(x^1/2)/1-x]
.where x= exp^(ħω/2)

E= -[δ/δβ ln(exp^-ħωβ / 1-exp^-ħωβ] = Eo - exp^-ħωβ / -1 + exp^-ħωβ
. where Eo= ħω/2

E= Eo + εħω
.where ε = 0.3ħω

I need find β in order to work out the temperatureof the system but stuck at this point. Really appreciate if some one can guide me in the right direction.

β = 1/(Kb*T)

where Kb is bolzmann's constant.

MaxL yes I know β= 1/kBT and have substituted β intentionally but what I am asking is how can i get a numerical value for β to work out the temperature? or do i need to? I'm just confused

thanks

When I did this, I got

$$E=\frac{\partial}{\partial \beta}\ln[z]=\frac{1}{2}\hbar\omega\coth[\beta\hbar\omega]$$

which looks pretty similar to what you have (you didn't take your exponentials to the hyperbolic functions).

Since you know that $E=\frac{1}{2}\hbar\omega+0.3\hbar\omega$, you can stick that into the equation and solve for $\beta$:

$$\frac{1}{2}\hbar\omega+0.3\hbar\omega=\frac{1}{2}\hbar\omega\coth[\beta\hbar\omega]$$

Sorry! I didn't mean to be condescending, I just read the question too fast.

Okay, so what you're ultimately after is an answer that looks like T=Teinstein*(Some number). It's pretty easy to show that Teinstein=ħωβ*T, so what you really need is to numerically determine ħωβ. You could get that using a calculator or mathematica to numerically solve an equation like the one jdwood just dropped.

Although, jdwood, I did not get that same equation. I think you may have made a chain rule mistake.

Hmm...I'll double check my work, but I'm pretty sure that when you take the derivative with respect to $\beta$, you get an exponential in the numerator and a $1-exp[\beta\hbar\omega]$ which can be reduced to the hyperbolic cotangent.

I'm pretty sure it is the same thing,

$$\coth[x]=\frac{\exp[x]+\exp[-x]}{\exp[x]-\exp[-x]}$$

and when you use the proper energy (the OP forgot a factor $\hbar\omega$ in his equation)

$$E=\frac{1}{2}\hbar\omega+\frac{\hbar\omega\exp[-\hbar\omega\beta]}{1-\exp[-\hbar\omega\beta]}=\frac{1}{2}\hbar\omega\left(\frac{1+\exp[-\hbar\omega\beta]}{1-\exp[-\hbar\omega\beta]}\right)$$

which is the same thing as the hyperbolic cotangent.

Okay, I double checked my work. My mistake!

Obi-wan, I hope that helps!

thank you for all your help. I've worked out the temperature to be ~ 0.6208ΘE which sounds reasonable

## 1. What is a quantum harmonic oscillator?

A quantum harmonic oscillator is a theoretical model that describes the behavior of a particle within a potential energy well. It is used to understand the dynamics of atoms, molecules, and other small-scale systems in quantum mechanics.

## 2. How is temperature defined in a quantum harmonic oscillator system?

In a quantum harmonic oscillator, temperature is defined as the average energy of the system divided by the Boltzmann constant. This allows us to relate the macroscopic concept of temperature to the microscopic behavior of particles in the system.

## 3. Can the temperature of a quantum harmonic oscillator be negative?

No, the temperature of a quantum harmonic oscillator cannot be negative. This is because temperature is a measure of the average energy of the system, and energy cannot be negative in quantum mechanics.

## 4. How does the temperature of a quantum harmonic oscillator affect its energy levels?

The energy levels of a quantum harmonic oscillator are directly related to its temperature. As the temperature increases, the energy levels become more closely spaced, meaning there is a higher probability of the oscillator occupying higher energy states.

## 5. How does the temperature of a quantum harmonic oscillator system change over time?

In a closed quantum harmonic oscillator system, the temperature will remain constant over time. However, if the system is in contact with a larger thermal reservoir, the temperature can change as energy is exchanged between the system and the reservoir.

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