Quantum Mechanics for Mathematicians
wheref 1 andf 2 are complex linear combinations of the powers of the anticom- muting variablesθj. For the details of the constru ...
corresponding to the action of an even or odd number of creation operators on | 0 〉F. This is because quadratic combinations of ...
and one creation operator, operators which annihilate| 0 〉F. Non-zero pairs of two creation operators act non-trivially on| 0 〉F ...
is the Hamiltonian for the classical fermionic oscillator. Quantizingh(see equa- tion 31.4) will give (−i) times the Hamiltonian ...
The quadratic Clifford algebra elements−^12 γjγkforj < ksatisfy the com- mutation relations ofso(4) =spin(4). These are expli ...
Spin(4) elements that act by unitary (for the Hermitian inner product 31.3) transformations on the spinor state space, but chang ...
Chapter 32 A Summary: Parallels Between Bosonic and Fermionic Quantization To summarize much of the material we have covered, it ...
Stone-von Neumann, Uniqueness ofh 2 d+1representation Uniqueness of Cliff(2d,C) representa- tion on spinors Mp(2d,R) double cove ...
Chapter 33 Supersymmetry, Some Simple Examples If one considers fermionic and bosonic quantum systems that each separately have ...
| 0 〉F. The Hamiltonian is H= 1 2 ~ω ∑d j=1 (aF†jaFj−aFjaF†j) = ∑d j=1 ( NFj− 1 2 ) ~ω whereNFjis the number operator for thej’t ...
0 ~ω 2 ~ω 3 ~ω Energy | 0 , 0 〉 | 1 , 0 〉 | 2 , 0 〉 | 3 , 0 〉 | 0 , 1 〉 | 1 , 1 〉 | 2 , 1 〉 Q+ Q+ Q+ Q− Q− Q− Figure 33.1: Energ ...
solutions to Q 1 | 0 〉= 0 or Q 2 | 0 〉= 0 The simplification here is much like what happens with the usual bosonic har- monic os ...
for the same reason as in the oscillator case: repeated factors ofaFora†Fvanish. Taking as the Hamiltonian the same square as be ...
In this simple quantum mechanical system, one can try and explicitly solve the equationQ 1 |ψ〉= 0. States can be written as two- ...
look familiar. This spaceHis well known to mathematicians, as the complex- valued differential forms onRd, often written Ω∗(Rd), ...
Chapter 34 The Pauli Equation and the Dirac Operator In chapter 33 we considered supersymmetric quantum mechanical systems where ...
on the spaceHB=L^2 (R^3 ) of square-integrable functions of the position coordinates. The Hamiltonian operator is H= 1 2 m |P|^2 ...
in terms of it). In this pseudo-classical theoryp 1 ξ 1 +p 2 ξ 2 +p 3 ξ 3 is the function generating a “supersymmetry”, Poisson ...
this equation becomes ((σ·p)^2 − 2 mE) ( ψ ̃ 1 (p) ψ ̃ 2 (p) ) = (|p|^2 − 2 mE) ( ψ ̃ 1 (p) ψ ̃ 2 (p) ) = 0 (34.4) and as in cha ...
and ̃u(a, 1 ) =e−ia·P so the Lie algebra representation is given by the usualPoperator. This action of the translations is easil ...
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