Quantum Mechanics for Mathematicians
one has the equality of distributions qδ(q−q′) =q′δ(q−q′) soδ(q−q′) is an eigenfunction ofQwith eigenvalueq′. The operatorsQandP ...
the momentum space representation, as opposed to the previous position space representation. By the Plancherel theorem (11.2) th ...
The resolution of the identity operator of equation 4.6 here is written 1 = ∫∞ −∞ |q〉〈q|dq= ∫∞ −∞ |k〉〈k|dk The transformation be ...
Theorem(Heisenberg uncertainty). 〈ψ|Q^2 |ψ〉 〈ψ|ψ〉 〈ψ|P^2 |ψ〉 〈ψ|ψ〉 ≥ 1 4 Proof.For any realλone has 〈(Q+iλP)ψ|(Q+iλP)ψ〉≥ 0 but, ...
In the Dirac notation one has ψ(qt,t) =〈qt|ψ(t)〉=〈qt|e−iHt|ψ(0)〉=〈qt|e−iHt ∫∞ −∞ |q 0 〉〈q 0 |ψ(0)〉dq 0 and the propagator can be ...
known as the “heat equation”. This equation models the way temperature diffuses in a medium, it also models the way probability ...
This is what one expects physically, sincep ′ mis the velocity corresponding to momentump′for a classical particle. Note that th ...
Definition(Retarded propagator).The retarded propagator is given by U+(t,qt−q 0 ) = { 0 t < 0 U(t,qt−q 0 ) t > 0 This can ...
C+ C− ω=−i Figure 12.2: Evaluatingθ(t) via contour integration. Fort > 0 , one instead closes the path usingC−in the lower h ...
as one expects since the delta-functionδ(ω) is the Fourier transform of 1 √ 2 π (θ(t) +θ(−t)) = 1 √ 2 π Returning to the propaga ...
12.7 Green’s functions and solutions to the Schr ̈o- dinger equations The method of Green’s functions provides solutionsψto diff ...
whereU+(t,q−q 0 ) is the retarded propagator given by equations 12.12 and 12.13. Since Dψ+(q,t) = (Dθ(t))ψ(q,t) +θ(t)Dψ(q,t) =iδ ...
12.8 For further reading The topics of this chapter are covered in every quantum mechanics textbook, with a discussion providing ...
Chapter 13 The Heisenberg group and the Schr ̈odinger Representation In our discussion of the free particle, we used just the ac ...
13.1 The Heisenberg Lie algebra In either the position or momentum space representation the operatorsPjand Qjsatisfy the relatio ...
Definition(Heisenberg Lie algebra).The Heisenberg Lie algebrah 2 d+1is the vector spaceR^2 d+1=R^2 d⊕Rwith the Lie bracket defin ...
group elements expressible as exponentials) by knowing the Lie bracket. For the full formula and a detailed proof, see chapter 5 ...
Definition(Schr ̈odinger representation, Lie algebra version).The Schr ̈odinger representation of the Heisenberg Lie algebrah 3 ...
as well as the same product in the opposite order, and then comparing the results. Note that, for the Schr ̈odinger representati ...
Note that all of this can easily be generalized to the case ofdspatial di- mensions, fordfinite, with the Heisenberg group nowH ...
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