Understanding Magnetisation for S=1/2: Exploring the Concept and Its Limitations

  • Thread starter Petar Mali
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In summary, the conversation discusses the magnetisation \sigma and the relationship between its value and the expectation value of the spin operators. It is shown that \sigma must be less than 1/2 and that the expectation value of the spin operators, when multiplied, must be between 0 and 1. The conversation also briefly mentions the definition of \sigma^{\pm} as the spin operators acting on the Pauli matrices \sigma_x and \sigma_y.
  • #1
Petar Mali
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If we have case

[tex]\sigma=\frac{1}{2}-\frac{1}{N}\sum_{\bf{k}}\langle\hat{S}^-\hat{S}^+\rangle_{\bf{k}}[/tex]

where [tex]\sigma[/tex] is magnetisation. How we know that [tex]\sigma[/tex] must be less than [tex]\frac{1}{2}[/tex]. Or why is

[tex]\frac{1}{N}\sum_{\bf{k}}\langle\hat{S}^-\hat{S}^+\rangle_{\bf{k}}>0[/tex]

Thanks for your answer.
 
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  • #2
Just look at the matrix elements of the spin raising and lowering operators.
Or alternatively, multiply out the spin operators to get

[tex]S^+ S^- = S^2 \sigma^+ \sigma^- = S^2 (\sigma_x^2 + \sigma_y^2 + 2\sigma_z)[/tex]

The [tex]\sigma_i^2[/tex] matrix is the identity, so its expectation value is 1. [tex]\sigma_z[/tex] has matrix elements of +1 and -1, so the quantity in the parentheses has to be between 0 and 4. S^2 is 1/4, so the result is between 0 and 1.
 
  • #3
daveyrocket said:
Just look at the matrix elements of the spin raising and lowering operators.
Or alternatively, multiply out the spin operators to get

[tex]S^+ S^- = S^2 \sigma^+ \sigma^- = S^2 (\sigma_x^2 + \sigma_y^2 + 2\sigma_z)[/tex]

The [tex]\sigma_i^2[/tex] matrix is the identity, so its expectation value is 1. [tex]\sigma_z[/tex] has matrix elements of +1 and -1, so the quantity in the parentheses has to be between 0 and 4. S^2 is 1/4, so the result is between 0 and 1.

You use if I see well

[tex]\hat{S}^+=S\hat{\sigma}^+[/tex]

[tex]\hat{S}^-=S\hat{\sigma}^-[/tex]

and how you define [tex]\hat{\sigma}^+[/tex] and [tex]\hat{\sigma}^-[/tex]?
 
  • #4
[tex]\sigma^{\pm} = \sigma_x \pm i\sigma_y[/tex]
 

1. What is magnetization for S=1/2?

Magnetization for S=1/2 refers to the magnetic moment per unit volume of a system with a spin quantum number (S) of 1/2. It is a measure of the alignment of the spin of the particles in the system.

2. How is magnetization for S=1/2 calculated?

Magnetization for S=1/2 is calculated by taking the average of the magnetic moments of all the particles in the system. This can be done by summing the magnetic moments and dividing by the total number of particles.

3. What is the significance of S=1/2 in magnetization?

S=1/2 is a common value for the spin quantum number in many types of particles, such as electrons. It is significant in magnetization because it helps determine the strength of the magnetic field and how the particles will align in response to an external magnetic field.

4. How does temperature affect magnetization for S=1/2?

At low temperatures, the spins of particles with S=1/2 tend to align and contribute to a higher magnetization. As temperature increases, thermal energy can disrupt the alignment and decrease the magnetization.

5. What are some real-world applications of magnetization for S=1/2?

Magnetization for S=1/2 is an important concept in the fields of materials science, condensed matter physics, and quantum mechanics. It has practical applications in technologies such as magnetic storage devices, magnetic sensors, and medical imaging machines.

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