Abstract
The distiguishability or not of quantum processes under an interchange operator \({\hat{P}}_{ij}\) of two particles and the interference phenomena of the wave function amplitudes, are fundamental issues of quantum theory. We will see that, for systems of identical particles, the invariance of the Hamiltonian under the action of the operator \({\hat{P}}_{ij}\) implies symmetry conditions on the wave functions that are symmetric under \({\hat{P}}_{ij}\) when it describes integer-spin particles (bosons), and antisymmetric, when it describes half-integer spin particles (fermions).
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Notes
- 1.
An observer in the center of mass, will see that two particles, approaching or distancing, move one opposite to the other before and after the scattering, independent of the scattering angle.
- 2.
Generally \(\varphi (\theta )=\varphi (-\theta )\) and \(\varphi (\pi -\theta )=\varphi (\theta -\pi )\).
- 3.
Which is equal to the probability of detecting an alpha particle in \(C_2\).
- 4.
W. Pauli, using field theory dealt with this issue.
- 5.
The matrix permanent develops like the matrix determinant, but without changes of sign.
- 6.
A detailed discussion on this issue, can be found in Richard P. Feynman, Robert Leighton and Matthew Sands, Lectures on Physics (Addison-Wesley, 1964).
- 7.
Eric Cornell and Carl Wieman, in 1995, have shown that cooling at \(0.17\,\upmu \text{ K}\), the rubidium gas undergoes to the Bose–Einstein condensate state.
- 8.
Pyotr Kapitsa, John Allen and Don Misener discovered that He\(^4\) passes to the superfluid state when the temperature reaches 2.17 K.
- 9.
When \(T=0\), it is common to call Fermi energy to Fermi level.
- 10.
Both representations are possible and one can establish the relation between them. This relation is a particular case of the transformation
$$\begin{aligned} |sm_ss_1s_2 \rangle&=\!\sum _{m_{s1}+m_{s2}=m_s}\!\!\!\!\!\! |s_1s_2m_{s1}m_{s2} \rangle \langle s_1s_2m_{s1}m_{s2} ||sm_ss_1s_2 \rangle \\ \!&=\!\sum _{m_{s1}+m_{s2}=m_s} \!\!\!\!\!\! C_{m_{s1},m_{s2},m_s}^{s_1,s_2,s} |s_1s_2m_{s1}m_{s2}\rangle . \nonumber \end{aligned}$$The coefficients \(C_{m_{s1},m_{s2},m_s}^{s_1,s_2,s}\) are known as the Clebsch-Gordan coefficients. The interested reader can find more details, for example, in A.R. Edmonds Angular Momentum in Quantum Mechanics, (Princeton University Press, Princeton, N. J. 1974).
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© 2012 Springer-Verlag Berlin Heidelberg
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Pereyra Padilla, P. (2012). Identical Particles, Bosons and Fermions . In: Fundamentals of Quantum Physics. Undergraduate Lecture Notes in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29378-8_13
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DOI: https://doi.org/10.1007/978-3-642-29378-8_13
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