Quantum Mechanics for Mathematicians
The Hamiltonian operator for the hydrogen atom acts trivially on theC^2 factor, so the only effect of the additional wavefunctio ...
Chapter 22 The Harmonic Oscillator In this chapter we’ll begin the study of the most important exactly solvable physical system, ...
22.1 The harmonic oscillator with one degree of freedom An even simpler case of a particle in a potential than the Coulomb poten ...
Instead of using two real coordinates to describe points in the phase space (and having to introduce a reality condition when us ...
with cn= ∫+∞ −∞ ψn(q)ψ(q,0)dq (note that theψnare real-valued). At later times, the wavefunction will be ψ(q,t) = ∑∞ n=0 cnψn(q) ...
Such an operator satisfies the commutation relations [N,a] = [a†a,a] =a†[a,a] + [a†,a]a=−a and [N,a†] =a† If|c〉is a normalized e ...
The rest of the energy eigenfunctions can be found by computing |n〉= a† √ n ··· a† √ 2 a† √ 1 | 0 〉= 1 √ n! (mω 2 ~ )n 2 ( q− ~ ...
plot gives the wavefunctions, which in this case are real and can be negative. The square of this function is what has an interp ...
state spaceH=F, whereF is the space of holomorphic functions (satisfying d dzψ= 0) onCwith finite norm in the inner product 〈ψ 1 ...
For completeness, assume〈n|ψ〉= 0 for alln. The expression for the|n〉 as Hermite polynomials times a Gaussian then implies that ...
a basis for the complexified dual phase spaceM⊗C. Note that these coordinates provide a decomposition M⊗C=C⊕C of the complexifie ...
Chapter 23 Coherent States and the Propagator for the Harmonic Oscillator In chapter 22 we found the energy eigenstates for the ...
what happens to this state under the Heisenberg group action. Elements of the complexified Heisenberg Lie algebrah 3 ⊗Ccan be wr ...
Definition(Coherent states).The coherent states inHare the states |α〉=D(α)| 0 〉=eαa †−αa | 0 〉 whereα∈C. Using the Baker-Campbel ...
Digression(Spin coherent states). One can perform a similar construction replacing the groupH 3 by the groupSU(2), and the state ...
In the Bargmann-Fock case, there is an analog of the distributional states |q〉, given by taking states that are eigenvectors for ...
This can be shown using |δw〉〈δw|= (∞ ∑ n=0 wn √ n! |n〉 )(∞ ∑ m=0 zm √ m! 〈m| ) as well as 1 = ∑∞ n=0 |n〉〈n| and the orthogonalit ...
We thus see that the Heisenberg group acts on annihilation and creation operators by shifting the operators by a constant. The H ...
with solution O(t) =eitHO(0)e−itH In the harmonic oscillator problem we can express other operators in terms of the annihilation ...
coordinatez(t) =√^12 ( √ ωq(t) +√iωp(t)) for the harmonic oscillator (see 22.1) withz(0) =α 0. Equations 23.3 and 23.9 can be us ...
«
9
10
11
12
13
14
15
16
17
18
»
Free download pdf