Download Lecture notes for FYS610 Many particle Quantum Mechanics

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Lecture notes for FYS610 Many particle Quantum Mechanics
Note 1, 13.1 2017
Additions and comments to Quantum
by Matthew D. Schwartz (2014)
Field Theory and the Standard Model
Canonical quantization
Quantum Mechanics, or more precisely Quantum Field Theory, is presumably a fundamental theory of nature, and as such cannot be derived from some other physical theory.
But in practice, we are often in the situation that we have a classical theory which describes physical phenomena well, and want to find a quantum theory which yields the
same observable predictions in the classical limit (in the simplest cases h̄ → 0). A procedure to derive such a quantum theory from a corresponding classical theory is called
a quantization procedure.
The best known such quantization procedure is known as canonical quantization, and
was discovered already by Dirac in his PhD thesis from 1926. This procedure works
best for unconstrained systems defined in Cartesian coordinates in Newtonian or specialrelativistic space-time. It is much harder, or impossible, to apply for constrained systems, including gauge field theories and systems defined in a curved space-time. Canonical quantization is not the only quantization procedure available. The most important
alternative is path integral quantization, which allows for the successful quantization
of gauge field theories (see S, ch. 14). These different quantization procedures yield
the same well known quantum theory in the case of unconstrained Newtonian systems.
In particular, they yield the important Ehrenfest’s theorem: Quantum mechanical
expectations values of dynamical observables obey the classical equations of motion.
Canonical quantization takes as its starting point a construction known as the Poisson bracket. We consider a classical mechanical system with N degrees of freedom
with unconstrained coordinates {qi }i=1...N and canonical momenta pi = ∂L/∂ q̇i , where
L({pi }{qi }, t) is the Lagrangian, and an dot denotes a time derivative. We know that
we can introduce the Hamiltonian as:
H=
N
X
pi q̇i − L .
(1.1)
i=1
Switching to the Hamiltonian formulation, with the pi ’s and qi ’s as independent variables, we have the Hamiltonian equations of motion:
ṗi = −
∂H
,
∂qi
q̇i =
1
∂H
.
∂pi
(1.2)
If A({pi }{qi }, t) and B({pi }{qi }, t) are two functions depending these coordinates, momenta and time, t, the Poisson bracket between A and B is defined as:
{A, B}P B =
N X
∂A ∂B
i
∂A ∂B
−
∂qi ∂pi
∂pi ∂qi
(1.3)
It is easily verified that the Poisson brackets satisfy the following identities, which are
defining a non-commutative algebra (c and d are constants):
{A, A}P B = 0
(1.4a)
{A, B}P B = −{B, A}P B
(antisymmetry)
(1.4b)
(linearity)
(1.4c)
{AB, C}P B = {A, B}P B C + A{B, C}P B (distributivity)
(1.4d)
{cA + dB, C}P B = c{A, C}P B + d{B, C}P B
In addition they satisfy a fifth identity, called the Jacobi identity, which require a much
more involved proof (see G, sect. 9.5):
{A, {B, C}P B }P B + {B, {C, A}P B }P B + {C, {A, B}P B }P B = 0 .
(1.5)
Eqs. (1.4-5) means that the dynamical variables A, B and C satisfy a non-associative
Lie algebra, with the Poisson bracket as a “multiplication” rule.
The remarkable feature of this Lie algebra, discovered by Dirac, is that it is identical to
the commutator algebra of the corresponding quantum mechanical operators. Thus, if
Â, B̂ and Ĉ are operators corresponding to the classical variables A, B and C, then these
operators automatically satisfies the same Lie algebra, just with the Poisson bracket
replaced by the commutator (see BJ [2.120]):
c{A, B}P B −→ [Â, B̂] = ÂB̂ − B̂ Â .
(1.6)
Here c is some constant, which so far is undetermined. Once this has been done, we
have the rule for canonical quantization. In the case of commutators it is trivial to
verify the Jacobi identity.
To find c let us consider the Poisson brackets of the basic dynamical variables, qi and
pj . Since they are independent variables, we have from the definition (1.3):
{pi , pj }P B = {qi , qj }P B = 0 ,
{qi , pj }P B =
N
X
(1.7a)
(δik δjk − 0) = δij .
(1.7b)
k=1
To compare with the corresponding commutator relation, we first observe that for any
operator Ô which only depend on the coordinates q1 . . . qn (and possibly t) we have in
coordinate representation with pi = −ih̄ ∂q∂ i , for an arbitrary wave function ψ(r):
[p̂i , Ô]ψ = −ih̄
∂(Ôψ)
∂ψ
− Ô
∂qi
∂qi
2
!
= −ih̄
∂ Ô
ψ.
∂qi
(1.8)
Since this is true for any ψ, it must be true as an operator identity. In the particular
case Ô = q̂j , we find:
[q̂i , p̂j ] = ih̄δij .
(1.9)
For this to be consistent with the correspondence (1.6), we must have the canonical
quantization rule:
1
{A, B}P B −→
[Â, B̂] .
(1.10)
ih̄
To see that this rule indeed gives the right dynamics, we verify that it gives rise to the
correct quantal equation of motion. If A is a classical dynamical variable as above, we
find its time evolution from the Hamiltonian equations of motion, eq. (1.2), as:
N
dA X
=
dt
i=1
∂A dqi
∂A dpi
+
∂qi dt
∂pi dt
= {A, H}P B +
∂A
∂t
−→
N
∂A X
+
=
∂t
i=1
∂A ∂H
∂A ∂H
−
∂qi ∂pi
∂pi ∂qi
+
∂A
∂t
(1.11)
1
∂ Â
[Â, Ĥ] +
.
ih̄
∂t
Thus canonical quantization yields Heisenberg’s equation of motion for  (see BJ
[2.113]).
We can now also prove Ehrenfest’s theorem. For simplicity, we shall assume that H has
the simple standard form:
N
X
p2j
H=
+ V(q1 . . . qN ) .
2mj
j=1
We find, using the commutator version of eqs. (1.3d) and (1.8):
[q̂i , H] =
1
p̂i
1
[q̂i , p̂2i ] =
([q̂i , p̂i ]p̂i + p̂i [q̂i , p̂i ]) = ih̄
,
2mi
2mi
mi
[p̂i , Ĥ] = [p̂i , V̂] =
h̄ ∂ V̂
h̄ ∂ Ĥ
=
.
i ∂qi
i ∂qi
We the have from the Heisenberg equations of motion, eq. (1.11):
mi
dp̂i
1
∂ Ĥ
= [p̂i , Ĥ] = −
.
dt
ih̄
∂qi
dq̂i
mi
=
[q̂i , Ĥ] = pi ,
dt
ih̄
This is nothing but the quantum mechanical Newtonian equations of motion, eqs. (1.2).
Taking the expectation value of these equations in an arbitrary state, we arrive at
Ehrenfest’s theorem.
References:
BJ) B. H. Bransden and C. J. Joachain: Physics of Atoms and Molecules (1983), ch.
2.1-3.
G) H. Goldstein, C. Poole and J. Safko: Classical Mechanics (3 ed. 2002), ch. 8.1,
9.1,5-6.
S) M. D. Schwartz: Quantum Field Theory and the Standard Model (2014).
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