Phase space representation of quantum dynamics - Boston University ...
Phase space representation of quantum dynamics. Lecture notes. Boulder Summer School, 2013 Anatoli Polkovnikov1 1
Department of Physics, Boston University, Boston, MA 02215
Contents I. Quick summary of classical Hamiltonian dynamics in phase-space. II. Quantum systems in first and second quantized forms. Coherent states. III. Wigner-Weyl quantization
1 4 6
A. Coordinate-Momentum representation
6
B. Coherent state representation.
11
C. Coordinate-momentum versus coherent state representations.
15
D. Spin systems.
16
IV. Quantum dynamics in phase space.
18
A. von Neumann’s equation in phase space representation. Truncated Wigner approximation
18
1. Single particle in a harmonic potential.
21
2. Collapse (and revival) of a coherent state
24
3. Spin dynamics in a linearly changing magnetic field: multi-level Landau-Zener problem.
26
V. Path integral derivation.
27
References
34
I. QUICK SUMMARY OF CLASSICAL HAMILTONIAN DYNAMICS IN PHASE-SPACE.
We will generally deal with Hamiltonian systems, which are defined by specifying a set of canonical variables pj , qj satisfying canonical relations {pi , qj } = δij ,
(1)
where {. . . } denotes the Poisson bracket. {A(~ p, ~q), B(~ p, ~q)} =
X ∂A ∂B ∂B ∂A − = BΛA, ∂pj ∂qj ∂pj ∂qj j
(2)
2 where ← − − X ← ∂ ∂ ∂ ∂ − Λ= ∂pj ∂qj ∂qj ∂pj j
is the sympectic skew symmetric operator. It is easy to check that any orthogonal transformation Q = R(λ)q, P = R(λ)p
(3)
preserves both the Poisson brackets and the symplectic operator. A general class of transformations which preserve the Poisson brackets are known as canonical transformations and can be expressed trhough the generating functions (1). It is easy to check that infinitesimal canonical transformations can be generated by gauge potentials ∂A(λ, p~, ~q) δλ, ∂pj ∂A(λ, p~, ~q) pj (λ + δλ) = pj (λ) + δλ, ∂qj
qj (λ + δλ) = qj (λ) −
(4) (5)
where λ parametrizes the canonical transformation and the gauge potential A is some function of canonical variables and parameters. Then up to the terms of the order of δλ2 the transformation above preserves the Poisson brackets {pi (λ + δλ), qj (λ + δλ)} = δij + δλ
∂2A ∂2A − ∂pj ∂qi ∂pj ∂qi
+ O(δλ2 ) = δij + O(δλ2 ).
(6)
Exercises. ~ is the momentum operator: A ~ ~ (~q, p~) = (i) Show that the generator of translations ~q(X) = ~q0 − X X ~ as a three component parameter ~λ. Note that the number of particles (and p~. You need to treat X thus phase space dimension) can be much higher than three. (ii) Show that the generator of the rotations around z-axis: qx (θ) = cos(θ)qx0 − sin(θ)qy0 , qy (θ) = cos(θ)qy0 + sin(θ)qx0 ,
px (θ) = cos(θ)px0 − sin(θ)py0 , py (θ) = cos(θ)py0 + sin(θ)px0 , is the angular momentum operator: Aθ = px qy − py qx . (iii) Find the gauge potential Aλ corresponding to the orthogonal transformation (3). Hamiltonian dynamics is a particular canonical transformation parametrized by time ∂qj ∂H ∂pj ∂H = {H, qj } = , = {H, pj } = − ∂t ∂pj ∂t ∂qj
(7)
3 Clearly these Hamiltonian equations are equivalent to Eqs. (5) with the convention At = −H. One can extend canonical transformations to the complex variables. Instead of doing this in all generality we will focus on particular phase space variables which are complex wave amplitudes. E.g. for Harmonic oscillators for each normal mode with the Hamiltonian Hk =
p2k mωk2 2 + q 2m 2 k
we can define new linear combinations r r mωk ∗ 1 pk = i (ak − ak ), qk = (ak + a∗k ) 2 2mωk
(8)
(9)
or equivalently a∗k
1 =√ 2
qk
√
√ i i 1 qk mωk + √ mωk − √ pk , a k = √ pk . mωk mωk 2
(10)
Let us now compute the Poisson brackets of the complex wave amplitudes {ak , ak } = {a∗k , a∗k } = 0, {ak , a∗k } = i.
(11)
To avoid dealing with the imaginary Poisson brackets it is convenient to introduce new coherent state Poisson brackets {A, B}c =
X ∂A ∂B ∂B ∂A − = AΛc B, ∂ak ∂a∗k ∂ak ∂a∗k
(12)
k
where Λc =
− ← − X ← ∂ ∂ ∂ ∂ − . ∂ak ∂a∗k ∂a∗k ∂ak
(13)
k
As for the coordinate momentum case, the coherent symplectic operator Λc is preserved under the canonical transformations. From this definition it is immediately clear that {ak , a∗q }c = δkq .
(14)
Comparing this relation with Eq. (11) we see that standard and coherent Poisson brackets differ by the factor of i: {. . . } = i{. . . }c .
(15)
Exercise. Check that any unitary transformation a ˜k = Uk,k0 a0k , where U is a unitary matrix, preserves the coherent state Poisson bracket, i.e. {˜ ak , a ˜∗q }c = δk,q . Verify that the Bogoliubov transformation γk = cosh(θk )ak + sinh(θk )a∗−k , γk∗ = cosh(θk )a∗k + sinh(θk )a−k ,
(16)
4 with θk = θ−k also preserves the coherent state Poisson bracket, i.e. ∗ ∗ {γk , γ−k }c = {γk , γ−k }c = 0, {γk , γk∗ }c = {γ−k , γ−k }c = 1.
(17)
Let us write the Hamiltonian equations of motion for the new coherent variables. Using that ∂A ∂A dA = − {A, H} = − i{A, H}c dt ∂t ∂t
(18)
and using that our variables do not explicitly depend on time (such dependence would amount to going to a moving frame, which we will not consider here) we find i
dak ∂H da∗k ∂H , i = {ak , H}c = = {a∗k , H}c = − dt ∂a∗k dt ∂ak
(19)
These equations are also known as Gross-Pitaevskii equations. Note that these equations are arbitrary for arbitrary Hamiltonians and not restricted to Harmonic systems. And finally let us write down the Liouville equations of motion for the probability distribution ρ(q, p, t) or ρ(a, a∗ , t). The latter just express incompressibility of the probability fllow, which directly follows conservation of the phase space volume dΓ = dqdp or dΓ = dada∗ for arbitrary canonical transformations including time evolution and from the conservation of the total probability ρdΓ: 0=
∂ρ ∂ρ dρ = − {ρ, H} = − i{ρ, H}c , dt ∂t ∂t
(20)
or equivalently ∂ρ ∂ρ = {ρ, H}, i = −{ρ, H}c ∂t ∂t
(21)
II. QUANTUM SYSTEMS IN FIRST AND SECOND QUANTIZED FORMS. COHERENT STATES.
Now we move to quantum systems. As for the classical systems let us first define the language. We will use two different representations of the operators using either coordinate-momentum (first quantized picture) or creation-annihilation operators (second quantized picture). In the second quantized form we will be only considering bosons because finding semiclassical limit for fermions is still an open question. These phase space variables satisfy canonical commutation relations: [ˆ qi , pˆj ] = i~δij , [ˆ ai , a ˆ†j ] = δij
(22)
Throughout these notes we introduce “hat”-notations for the operators to avoid confusion with the phase space variables. From this relations it is clear that in the classical limit the commutator
5 should reduce to the coherent state Poisson bracket. As in the classical systems any Unitary transformation of the canonical variables preserves their commutation relations. Since we will be always keeping in mind the classical limit we will be predominantly working in the Heisenberg representation where the operators are time dependent and satisfy canonical equations of motion dˆ qi pi ˆ i~ dˆ ˆ = [ˆ qi , H], = [ˆ pi , H], dt dt a†i dˆ ai ˆ i~ dˆ ˆ i~ = [ˆ ai , H], = [ˆ a†i , H]. dt dt i~
(23) (24)
As in the classical case these equations can be thought of as continuous canonical transformations parametrized by time. Next let us define representation of these operators. For canonical coordinate and momentum the natural representation, which is most often used in literature is coordinate, where qˆj → xj , pˆj = −i~
∂ ∂xj
(25)
This representation is realized using coordinate eigenstates |~xi = |x1 , x2 , . . . , xM i such that any state |ψi is written as Z |ψi =
D~x ψ(~x)|~xi.
(26)
Here M denotes the total number of independent coordinate components, e.g. in the threedimensional space M is equal to three times the number of particles. In a similar fashion the natural representation for creation and annihilation operators is given by coherent states: a ˆj → αj , a ˆ†j → −
∂ ∂αj
(27)
Clearly in this form the creation and annihilation operators satisfy canonical commutation relations (22). Coherent states can be created from the vacuum state by exponentiating the creation operator: |α1 , α2 , . . . αM i =
M Y
e−|αj |
2 /2
†
eαj aj |0i,
(28)
j=1
where |0i is the global particle vacuum annihilated by all operators a ˆj 1 . One can check that these 1
Note that there is a sign mismatch between a ˆ† |αi = ∂α |αi and the representation (27). This is because the derivative operator acting on the basis vector is opposite in sign to the derivative operator acting on the wave R function |ψi = dαψ(α)|αi.
6 coherent states are properly normalized: Z
DαDα∗ hα1 , α2 , . . . αM |α1 , α2 , . . . αM i = 1,
(29)
where we use the integration measure dαdα∗ = d
Department of Physics, Boston University, Boston, MA 02215
Contents I. Quick summary of classical Hamiltonian dynamics in phase-space. II. Quantum systems in first and second quantized forms. Coherent states. III. Wigner-Weyl quantization
1 4 6
A. Coordinate-Momentum representation
6
B. Coherent state representation.
11
C. Coordinate-momentum versus coherent state representations.
15
D. Spin systems.
16
IV. Quantum dynamics in phase space.
18
A. von Neumann’s equation in phase space representation. Truncated Wigner approximation
18
1. Single particle in a harmonic potential.
21
2. Collapse (and revival) of a coherent state
24
3. Spin dynamics in a linearly changing magnetic field: multi-level Landau-Zener problem.
26
V. Path integral derivation.
27
References
34
I. QUICK SUMMARY OF CLASSICAL HAMILTONIAN DYNAMICS IN PHASE-SPACE.
We will generally deal with Hamiltonian systems, which are defined by specifying a set of canonical variables pj , qj satisfying canonical relations {pi , qj } = δij ,
(1)
where {. . . } denotes the Poisson bracket. {A(~ p, ~q), B(~ p, ~q)} =
X ∂A ∂B ∂B ∂A − = BΛA, ∂pj ∂qj ∂pj ∂qj j
(2)
2 where ← − − X ← ∂ ∂ ∂ ∂ − Λ= ∂pj ∂qj ∂qj ∂pj j
is the sympectic skew symmetric operator. It is easy to check that any orthogonal transformation Q = R(λ)q, P = R(λ)p
(3)
preserves both the Poisson brackets and the symplectic operator. A general class of transformations which preserve the Poisson brackets are known as canonical transformations and can be expressed trhough the generating functions (1). It is easy to check that infinitesimal canonical transformations can be generated by gauge potentials ∂A(λ, p~, ~q) δλ, ∂pj ∂A(λ, p~, ~q) pj (λ + δλ) = pj (λ) + δλ, ∂qj
qj (λ + δλ) = qj (λ) −
(4) (5)
where λ parametrizes the canonical transformation and the gauge potential A is some function of canonical variables and parameters. Then up to the terms of the order of δλ2 the transformation above preserves the Poisson brackets {pi (λ + δλ), qj (λ + δλ)} = δij + δλ
∂2A ∂2A − ∂pj ∂qi ∂pj ∂qi
+ O(δλ2 ) = δij + O(δλ2 ).
(6)
Exercises. ~ is the momentum operator: A ~ ~ (~q, p~) = (i) Show that the generator of translations ~q(X) = ~q0 − X X ~ as a three component parameter ~λ. Note that the number of particles (and p~. You need to treat X thus phase space dimension) can be much higher than three. (ii) Show that the generator of the rotations around z-axis: qx (θ) = cos(θ)qx0 − sin(θ)qy0 , qy (θ) = cos(θ)qy0 + sin(θ)qx0 ,
px (θ) = cos(θ)px0 − sin(θ)py0 , py (θ) = cos(θ)py0 + sin(θ)px0 , is the angular momentum operator: Aθ = px qy − py qx . (iii) Find the gauge potential Aλ corresponding to the orthogonal transformation (3). Hamiltonian dynamics is a particular canonical transformation parametrized by time ∂qj ∂H ∂pj ∂H = {H, qj } = , = {H, pj } = − ∂t ∂pj ∂t ∂qj
(7)
3 Clearly these Hamiltonian equations are equivalent to Eqs. (5) with the convention At = −H. One can extend canonical transformations to the complex variables. Instead of doing this in all generality we will focus on particular phase space variables which are complex wave amplitudes. E.g. for Harmonic oscillators for each normal mode with the Hamiltonian Hk =
p2k mωk2 2 + q 2m 2 k
we can define new linear combinations r r mωk ∗ 1 pk = i (ak − ak ), qk = (ak + a∗k ) 2 2mωk
(8)
(9)
or equivalently a∗k
1 =√ 2
qk
√
√ i i 1 qk mωk + √ mωk − √ pk , a k = √ pk . mωk mωk 2
(10)
Let us now compute the Poisson brackets of the complex wave amplitudes {ak , ak } = {a∗k , a∗k } = 0, {ak , a∗k } = i.
(11)
To avoid dealing with the imaginary Poisson brackets it is convenient to introduce new coherent state Poisson brackets {A, B}c =
X ∂A ∂B ∂B ∂A − = AΛc B, ∂ak ∂a∗k ∂ak ∂a∗k
(12)
k
where Λc =
− ← − X ← ∂ ∂ ∂ ∂ − . ∂ak ∂a∗k ∂a∗k ∂ak
(13)
k
As for the coordinate momentum case, the coherent symplectic operator Λc is preserved under the canonical transformations. From this definition it is immediately clear that {ak , a∗q }c = δkq .
(14)
Comparing this relation with Eq. (11) we see that standard and coherent Poisson brackets differ by the factor of i: {. . . } = i{. . . }c .
(15)
Exercise. Check that any unitary transformation a ˜k = Uk,k0 a0k , where U is a unitary matrix, preserves the coherent state Poisson bracket, i.e. {˜ ak , a ˜∗q }c = δk,q . Verify that the Bogoliubov transformation γk = cosh(θk )ak + sinh(θk )a∗−k , γk∗ = cosh(θk )a∗k + sinh(θk )a−k ,
(16)
4 with θk = θ−k also preserves the coherent state Poisson bracket, i.e. ∗ ∗ {γk , γ−k }c = {γk , γ−k }c = 0, {γk , γk∗ }c = {γ−k , γ−k }c = 1.
(17)
Let us write the Hamiltonian equations of motion for the new coherent variables. Using that ∂A ∂A dA = − {A, H} = − i{A, H}c dt ∂t ∂t
(18)
and using that our variables do not explicitly depend on time (such dependence would amount to going to a moving frame, which we will not consider here) we find i
dak ∂H da∗k ∂H , i = {ak , H}c = = {a∗k , H}c = − dt ∂a∗k dt ∂ak
(19)
These equations are also known as Gross-Pitaevskii equations. Note that these equations are arbitrary for arbitrary Hamiltonians and not restricted to Harmonic systems. And finally let us write down the Liouville equations of motion for the probability distribution ρ(q, p, t) or ρ(a, a∗ , t). The latter just express incompressibility of the probability fllow, which directly follows conservation of the phase space volume dΓ = dqdp or dΓ = dada∗ for arbitrary canonical transformations including time evolution and from the conservation of the total probability ρdΓ: 0=
∂ρ ∂ρ dρ = − {ρ, H} = − i{ρ, H}c , dt ∂t ∂t
(20)
or equivalently ∂ρ ∂ρ = {ρ, H}, i = −{ρ, H}c ∂t ∂t
(21)
II. QUANTUM SYSTEMS IN FIRST AND SECOND QUANTIZED FORMS. COHERENT STATES.
Now we move to quantum systems. As for the classical systems let us first define the language. We will use two different representations of the operators using either coordinate-momentum (first quantized picture) or creation-annihilation operators (second quantized picture). In the second quantized form we will be only considering bosons because finding semiclassical limit for fermions is still an open question. These phase space variables satisfy canonical commutation relations: [ˆ qi , pˆj ] = i~δij , [ˆ ai , a ˆ†j ] = δij
(22)
Throughout these notes we introduce “hat”-notations for the operators to avoid confusion with the phase space variables. From this relations it is clear that in the classical limit the commutator
5 should reduce to the coherent state Poisson bracket. As in the classical systems any Unitary transformation of the canonical variables preserves their commutation relations. Since we will be always keeping in mind the classical limit we will be predominantly working in the Heisenberg representation where the operators are time dependent and satisfy canonical equations of motion dˆ qi pi ˆ i~ dˆ ˆ = [ˆ qi , H], = [ˆ pi , H], dt dt a†i dˆ ai ˆ i~ dˆ ˆ i~ = [ˆ ai , H], = [ˆ a†i , H]. dt dt i~
(23) (24)
As in the classical case these equations can be thought of as continuous canonical transformations parametrized by time. Next let us define representation of these operators. For canonical coordinate and momentum the natural representation, which is most often used in literature is coordinate, where qˆj → xj , pˆj = −i~
∂ ∂xj
(25)
This representation is realized using coordinate eigenstates |~xi = |x1 , x2 , . . . , xM i such that any state |ψi is written as Z |ψi =
D~x ψ(~x)|~xi.
(26)
Here M denotes the total number of independent coordinate components, e.g. in the threedimensional space M is equal to three times the number of particles. In a similar fashion the natural representation for creation and annihilation operators is given by coherent states: a ˆj → αj , a ˆ†j → −
∂ ∂αj
(27)
Clearly in this form the creation and annihilation operators satisfy canonical commutation relations (22). Coherent states can be created from the vacuum state by exponentiating the creation operator: |α1 , α2 , . . . αM i =
M Y
e−|αj |
2 /2
†
eαj aj |0i,
(28)
j=1
where |0i is the global particle vacuum annihilated by all operators a ˆj 1 . One can check that these 1
Note that there is a sign mismatch between a ˆ† |αi = ∂α |αi and the representation (27). This is because the derivative operator acting on the basis vector is opposite in sign to the derivative operator acting on the wave R function |ψi = dαψ(α)|αi.
6 coherent states are properly normalized: Z
DαDα∗ hα1 , α2 , . . . αM |α1 , α2 , . . . αM i = 1,
(29)
where we use the integration measure dαdα∗ = d
