Average energy of a harmonic oscillator

[Rajini]

Hello PF members,
Is there some good book, which contain the derivation of average energy of a harmonic oscillator at temperature T. I want to derive from Planck's distribution (PD) function (<n>=(exp(##\hbar\omega/kT##)-1)##^{-1}##)...to get the following relation:
energy E= (##\hbar\omega##/2)+(##\hbar\omega##/PD). I referred to some books/www..they mainly refer E= ##\hbar\omega##(n+(1/2))...stating n as 0,1,2,3,etc...
But i think this n is simply the PD..
Can some one help me..
thanks

 

[azntoon]

in advance.There is no single book that covers all of the topics you're asking about, but there are several good resources available. A great starting point would be the textbook by Shankar, "Principles of Quantum Mechanics" (2nd edition). This book contains a thorough discussion of the harmonic oscillator and its properties, including the average energy at a given temperature. Additionally, Shankar provides a derivation of the Planck distribution function and its application to the harmonic oscillator. Another useful resource is the textbook by Sakurai, "Modern Quantum Mechanics" (2nd edition). This book also contains a detailed discussion of the harmonic oscillator and its properties, including the energy at a given temperature. In this book, the derivation of the Planck distribution is provided, as well as a discussion of the relation between the energy and the Planck distribution. Finally, you may want to check out the online lecture notes by Griffiths, "Introduction to Quantum Mechanics" (2nd edition). These lecture notes provide a comprehensive overview of the harmonic oscillator, including a derivation of the Planck distribution and the corresponding relation between the energy and the Planck distribution. I hope this information is helpful!

 

[Entropia]

Hello there,

The average energy of a harmonic oscillator at temperature T can indeed be derived from the Planck distribution function. The expression for the average energy is given by <E> = (1/ PD) * ∑n E(n) * P(n), where E(n) is the energy of the oscillator in the nth energy level and P(n) is the probability of the oscillator being in that energy level.

Using the Planck distribution function, we can write the probability as P(n) = (exp( -E(n)/kT) / (1 - exp(-E(n)/kT)). Substituting this in the expression for average energy, we get <E> = (1/ PD) * ∑n E(n) * (exp( -E(n)/kT) / (1 - exp(-E(n)/kT)).

Now, for a harmonic oscillator, the energy levels are given by E(n) = (n + 1/2) * ℏω. Substituting this in the above expression and simplifying, we get <E> = (ℏω/2) + (ℏω/PD). This is the relation that you were looking for.

I hope this helps. If you need more clarification or assistance, please feel free to ask. Also, there are many good books that cover this topic, such as "Introduction to Quantum Mechanics" by David J. Griffiths and "Quantum Mechanics" by Leonard I. Schiff. I suggest you refer to these for a more detailed derivation and understanding of the average energy of a harmonic oscillator.