Statistical mechanics: Particles with spin

In summary, this conversation discusses the energy of a system of N particles in a magnetic field, with each particle having a spin-up or spin-down state. The canonic partition function and the chance of a particle being in the spin-up state are calculated for N = 1. For any N, the number of microstates, Helmholtz free energy, and average energy per particle are also calculated. The number of possible microstates is 2^N and the calculation of Z is left as a summation.
  • #1
SoggyBottoms
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Homework Statement



We have N particles, each of which can either be spin-up ([itex]s_i = 1[/itex]) or spin-down ([itex]s_i = -1[/itex]) with [itex]i = 1, 2, 3...N[/itex]. The particles are in fixed position, don't interact and because they are in a magnetic field with strength B, the energy of the system is given by:

[tex]E(s_1, ..., s_n) = -mB \sum_{i=1}^{N} s_i[/tex]

with m > 0 the magnetic moment of the particles. The temperature is T.

a) Calculate the canonic partition function for N = 1 and the chance that this particle is in spin-up state [itex]P_+[/itex].

b) For any N, calculate the number of microstates [itex]\Omega(N)[/itex], the Helmholtz free energy F(N,T) and the average energy per particle U(N, T)/N

The Attempt at a Solution



a) [tex]Z_1 = e^{-\beta m B} + e^{\beta m B} = 2 \cosh{\beta m B}[/tex]
[tex]P_+ = \frac{e^{-\beta m B}}{2 \cosh{\beta m B}}[/tex]

b) The number of possible microstates is [itex]\Omega(N) = 2^N[/itex], correct?

I know that [itex]U = -\frac{\partial \ln Z}{\partial \beta}[/itex], but I'm not sure how to calculate Z here.
 
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  • #2
leave Z as the summation Z = Ʃ e-Eiβ where β = 1/KBTso ∂ln(Z)/dβ = (1/Z)(∂Z/∂β) = [-EiƩe-Eiβ]/Zi think b) is supposed to be (Z1)N sorry yeah your b) is right
 

Related to Statistical mechanics: Particles with spin

1. What is statistical mechanics?

Statistical mechanics is a branch of physics that uses statistical methods to study the behavior of large systems of particles, such as atoms and molecules. It seeks to understand the macroscopic properties of these systems by analyzing the microscopic behavior of their individual particles.

2. What is the difference between classical and quantum statistical mechanics?

Classical statistical mechanics describes systems made up of particles with no intrinsic spin, such as macroscopic objects. Quantum statistical mechanics, on the other hand, takes into account the intrinsic spin of particles, which is a fundamental property of quantum mechanics.

3. How does statistical mechanics explain thermodynamics?

Statistical mechanics provides a microscopic understanding of the laws of thermodynamics, which describe the behavior of macroscopic systems. By studying the statistical behavior of individual particles, it can explain macroscopic phenomena such as temperature, pressure, and energy transfer.

4. What is the significance of particles with spin in statistical mechanics?

Particles with spin play a crucial role in statistical mechanics because they exhibit quantum mechanical properties that cannot be explained by classical physics. The spin of a particle affects its energy levels and how it interacts with other particles, which is essential for understanding the behavior of complex systems.

5. What are some applications of statistical mechanics?

Statistical mechanics has a wide range of applications, including in condensed matter physics, astrophysics, and chemistry. It is used to study the behavior of gases, liquids, and solids, as well as complex systems such as biological molecules and the entire universe. It is also essential in the development of new materials and technologies.

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