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welshtill
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I am reading J.W.Negele and H.Orland's book "Quantum Many-Particle Systems". I don't know how one can derive equation (1.40) on page 6. The question is
For quantum many-body physics, suppose there are N particles. The hilbert space is
[tex]H_{N}=H\otimesH\otimes...H[/tex].
Its basis can be written as
[tex]\left|\alpha_{1}\alpha_{2}...\alpha_{N}\right)=\left|\alpha_{1}\right\rangle\otimes\left|\alpha_{2}\right\rangle\...\otimes\left|\alpha_{N}\right\rangle[/tex]
with closure relation
[tex]\sum_{\alpha_{1}\alpha_{2}...\alpha_{N}}\left|\alpha_{1}\alpha_{2}...\alpha_{N}\right)\left(\alpha_{1}\alpha_{2}...\alpha_{N}\right|=1[/tex]
Now introduce a symmetrization and antisymmetrization operator
[tex]P_{B,F}\psi(r_{1},r_{2}...r_{N})=\frac{1}{N!}\sum_{P}\varsigma^{P}\psi(r_{P1},r_{P2}...r_{PN})[/tex]
where [tex]\varsigma=1[/tex] for bosons and -1 for fermions. [tex]\sum_{P}[/tex] is sum of all permutations of coordinates.
Using this operator [tex]P_{B,F}[/tex] one can obtain the sub-hilbert space for fermions [tex]H_{F}[/tex] or for bosons [tex]H_{B}[/tex].
The basis in these two sub-space can be written as
[tex]P_{B,F}\left|\alpha_{1}\alpha_{2}...\alpha_{N}\right)[/tex]
My question is how one can derive the following equation
[tex]\sum_{\alpha_{1}\alpha_{2}...\alpha_{N}}P_{B,F}\left|\alpha_{1}\alpha_{2}...\alpha_{N}\right)\left(\alpha_{1}\alpha_{2}...\alpha_{N}\right|P_{B,F}=1[/tex]
For quantum many-body physics, suppose there are N particles. The hilbert space is
[tex]H_{N}=H\otimesH\otimes...H[/tex].
Its basis can be written as
[tex]\left|\alpha_{1}\alpha_{2}...\alpha_{N}\right)=\left|\alpha_{1}\right\rangle\otimes\left|\alpha_{2}\right\rangle\...\otimes\left|\alpha_{N}\right\rangle[/tex]
with closure relation
[tex]\sum_{\alpha_{1}\alpha_{2}...\alpha_{N}}\left|\alpha_{1}\alpha_{2}...\alpha_{N}\right)\left(\alpha_{1}\alpha_{2}...\alpha_{N}\right|=1[/tex]
Now introduce a symmetrization and antisymmetrization operator
[tex]P_{B,F}\psi(r_{1},r_{2}...r_{N})=\frac{1}{N!}\sum_{P}\varsigma^{P}\psi(r_{P1},r_{P2}...r_{PN})[/tex]
where [tex]\varsigma=1[/tex] for bosons and -1 for fermions. [tex]\sum_{P}[/tex] is sum of all permutations of coordinates.
Using this operator [tex]P_{B,F}[/tex] one can obtain the sub-hilbert space for fermions [tex]H_{F}[/tex] or for bosons [tex]H_{B}[/tex].
The basis in these two sub-space can be written as
[tex]P_{B,F}\left|\alpha_{1}\alpha_{2}...\alpha_{N}\right)[/tex]
My question is how one can derive the following equation
[tex]\sum_{\alpha_{1}\alpha_{2}...\alpha_{N}}P_{B,F}\left|\alpha_{1}\alpha_{2}...\alpha_{N}\right)\left(\alpha_{1}\alpha_{2}...\alpha_{N}\right|P_{B,F}=1[/tex]
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