secondquantization

# secondquantization - Phys 852 Quantum mechanics II Spring...

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Unformatted text preview: Phys 852, Quantum mechanics II, Spring 2008 Second Quantization: Non-Relativistic Quantum Field Theory 4/21/2008 Prof. Michael G. Moore, Michigan State University In standard one-body quantum mechanics, the state of the system | ψ ) is completely determined by the wavefunction φ ( vector r ), which is the state of the system projected onto the basis of position eigenstates {| vector r )} . The dynamical evolution of this wavefunction is governed by Schr¨odinger’s equation i planckover2pi1 d dt φ ( vector r ) = bracketleftbigg − planckover2pi1 2 2 m ∇ 2 r + V ( vector r ) bracketrightbigg φ ( vector r ) . (1) Standard two-body quantum-mechanics introduces a second particle, so that a complete basis becomes the joint position eigenstates {| vector r 1 ,vector r 2 )} . The state of the system is then completely determined by the two-body wavefunction φ ( vector r 1 ,vector r 1 ). The complete wave equation for this two-body wavefunction is i planckover2pi1 d dt φ ( vector r 1 ,vector r 2 ) = bracketleftbigg − planckover2pi1 2 2 m 1 ∇ 2 r 1 + V 1 ( vector r 1 ) bracketrightbigg φ ( vector r 1 ,vector r 2 ) + bracketleftbigg − planckover2pi1 2 2 m 2 ∇ 2 r 2 + V 2 ( vector r 2 ) bracketrightbigg φ ( vector r 1 ,vector r 2 ) + V 12 ( vector r 1 ,vector r 2 ) φ ( vector r 1 ,vector r 2 ) , (2) where V 1 is the potential seen by particle 1 in absence of particle 2, V 2 is the potential seen by 2 in the absence of 1, and V 12 is the interaction potential for particles 1 and 2. In the case of identical particles we have m 1 = m 2 , V 1 = V 2 , V 12 = V 21 , and φ ( vector r 2 ,vector r 2 ) = ± φ ( vector r 1 ,vector r 2 ) (+ for bosons and − for fermions). For identical particles we can generalize to an N-body system, by projecting the state of the system onto the joint position eigenstate bases {| vector r 1 ,...,vector r N )} . This leads to the N-body Schr¨odinger equation i planckover2pi1 d dt ψ ( vector r 1 ,...,vector r N ) = N summationdisplay j =1 bracketleftbigg − planckover2pi1 2 2 m ∇ 2 r j + V (1) ( vector r j ) bracketrightbigg φ ( vector r 1 ,...,vector r N ) + 1 2 summationdisplay j negationslash = i V (2) ( vector r i ,vector r j ) φ ( vector r 1 ,...,vector r N ) + 1 3! summationdisplay i negationslash = j negationslash = k V (3) ( vector r i ,vector r j ,vector r k ) φ ( vector r 1 ,... ,vector r N ) + ..., (3) where V ( n ) is the n-body interaction term. In many situations, it will only be necessary to keep the n = 1 and n = 2 terms, as three body interactions are often absent or negligible. This method of doing many-body quantum mechanics is correct, but it can be somewhat unwieldy and does not allow for states without a well-defined particle number....
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secondquantization - Phys 852 Quantum mechanics II Spring...

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