Download Fundamentals of quantum mechanics Quantum Theory of Light and Matter

Fundamentals
Fundamentals of quantum mechanics
Quantum Theory of Light and Matter
Quantum mechanics revision
Elementary description in terms of wavefunction ψ(x)
|ψ(x)|2 : probability measuring particle at position x
More generally: state vector |ψi (cf ~a)
P
Scalar product: ha|bi – in components i ai∗ bi .
Physical properties relate to Hermitian operators
Ô|ψi → |φi
Paul Eastham
Operator algebra defines theory
Position and momentum [q̂, p̂] = i~
Fundamentals
Fundamentals
Fundamentals of quantum mechanics
Fundamentals of quantum mechanics
Measurements associated with Hermitian operators Ô
Eigenstate of Ô|ei i = λi |ei i
Arbitrary state= superposition of eigenstates of Ô
Time evolution of state vector obeys
i~
∂
|ψi = Ĥ|ψ(t).
∂t
|ψi =
Usual Hamiltonian is classical energy with p, q → p̂, q̂
Ĥ =
p̂2
2m
X
i
ci |ei i.
Measurement gives λi with probability |ci |2 .
After measurement state is |ei i.
+ V (q̂).
[Ô1 , Ô2 ] 6= 0 ⇒ eigenstates not orthogonal.
measurement of O1 ⇒ superposition of e’states of O2 ⇒
uncertainty etc..
Origin of uncertainty relations
Origin of uncertainty relations
A general uncertainty relation
A general uncertainty relation
Many systems in some identical quantum state.
Measure the values for Â, usual variance of the resulting data
set
Same for
σA2 σB2
Useful if [A, B] = number, ic, same irrespective of state
⇒
σA2 σB2 = ha|aihb|bi
= ha|aihb|bi
≥ |ha|bi|2
= hψ|(ÂB̂)|ψi . . .
⇒ha|bi − hb|ai = hψ[A, B]|ψi.
σA2 = hψ|(Â − hAi)(Â − hA)|ψi = ha|ai
σB2
z = ha|bi = hψ|(Â − hAi)(B̂ − hBi)|ψi
= |z|2 ≥ (Imz)2 =
z −z
2i
∗ 2
≥ |ha|bi|2
= |z|2 ≥ (Imz)2 =
z − z∗
2i
=
2
c2
4
Origin of uncertainty relations
Heisenberg and Schrodinger Pictures
A general uncertainty relation
Heisenberg and Schrodinger Pictures
c
2
~
σp σq ≥
2
σA σB ≥
Generalization of Heisenberg uncertainty relation
About parallelism of eigenvectors;
[A, B] = ic type operators can bound max angle < 90◦
Remember this is a bound not an equality
“Minimum uncertainty states” saturate this bound
This is not a statement that  (or B̂) alone is “imprecise”
Asymmetrical uncertainty, e.g. σA2 < c 2 /4, “squeezed state”
Heisenberg and Schrodinger Pictures
Ladder operators for the harmonic oscillator
Operator approach to harmonic oscillator
Operator approach to harmonic oscillator
EM field equivalent to a set of harmonic oscillators ∴
Ĥ =
mω 2 2
p̂2
+
q̂ ,
2m
2
“Annihilation operator” â –
â =
r
mω
q̂ + i
2~
r
1
p̂,
2~mω
⇒
1
Ĥ = ~ω a† a +
2
Suppose eigenstate of a† a w/eval n = |ni.
√
â|ni ∝ |n − 1i = n|n − 1i
√
↠|ni ∝ |n + 1i = n + 1|ni.
Lower bound on H ⇒ some state â|0i = 0.
∴ discrete ladder of integer n ⇒ energy ~ω(n + 1/2).
”Number states” or “Fock states”
[q̂, p̂] = i~ ⇒ [â, ↠] = 1
Ladder operators for the harmonic oscillator
Ladder operators for the harmonic oscillator
Ladder operator proofs
Operator approach to harmonic oscillator
If ↠â|ni = n|ni, consider
↠â(â|ni)
= (â↠− 1)â|ni
([a, a† ] = 1)
†
= â(â â|ni) − â|ni
= (n − 1)(â|ni).
So â|ni is an eigenstate with eigenvalue |n − 1i:
â|ni = c|n − 1i.
and mod-squaring this gives normalization |c|2 = n.
Interested in position? Will need hq|ni.
1
q
d
â = √
+`
.
dq
2 `
So ground state obeys a|0i = 0 or
1
q
d
√
+`
u0 (q) = 0
dq
2 `
1/4
1
1
2
2
√ e−q /2` .
⇒ u0 (q) =
π
`
How do we get the excited state wavefunctions . . . ?