presentation qhjt 1final
TRANSCRIPT
-
8/13/2019 Presentation Qhjt 1final
1/24
Quantum Hamilton Jacobi Theory
April 15, 2013
Quantum Hamilton Jacobi Theory April 15, 2013 1 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
2/24
IntroductionClassical Canonical Transformations
Classical canonical transformations are useful for finding a suitable set ofdynamical variables for a particular problem.
qQ
pPH(q, p, t)K(Q,P, t)
Canonical transformations are induced by a generating function, F.
F can be a function of any two independent variables e.g. (q,Q), (q,P),(Q,p) and (p,P), corresponding to the different types of generatingfunction.
Quantum Hamilton Jacobi Theory April 15, 2013 2 / 24
http://goforward/http://find/http://goback/ -
8/13/2019 Presentation Qhjt 1final
3/24
IntroductionType-I Generating Functions
pi=
qiF1(q,Q, t)
Pi=
QiF1(q,Q, t)
K(Q,P, t) =H(q, p, t) +
tF1(q,Q, t)
Quantum Hamilton Jacobi Theory April 15, 2013 3 / 24
http://find/http://goback/ -
8/13/2019 Presentation Qhjt 1final
4/24
IntroductionHamilton-Jacobi Equation
If we manage to find a canonical transformation which gives us
K(Q,P,t)=0, then we get the Hamilton-Jacobi equation
0= H(q,F1
q , t) +
tF1(q,Q, t)
Quantum Hamilton Jacobi Theory April 15, 2013 4 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
5/24
Quantum Canonical Transformations
There also exist Quantum Canonical Transformations, which are similar totheir classical counterparts
q Q
p P
H(q, p, t)K(Q,P, t)
Quantum Hamilton Jacobi Theory April 15, 2013 5 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
6/24
Quantum Canonical TransformationsType-I QCT
For a type-I quantum canonical transformation we would have
pi=
qiF1(q,Q, t)
Pi=
QiF1(q,Q, t)
K(Q,P, t) =H(q, p, t) +
t
F1(q,Q, t)
Quantum Hamilton Jacobi Theory April 15, 2013 6 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
7/24
Quantum Canonical TransformationsWell-ordering
However the quantum generating function may contain noncommutingoperators, so we must enforce well-ordering: operators with upper-case
letters should always be to the right of operators with lower-case letters, or
F1(q,Q, t) =
f(q, t)g(Q, t)
From here we shall refer to F1(q,Q, t) as W(q,Q, t).
Quantum Hamilton Jacobi Theory April 15, 2013 7 / 24
http://goforward/http://find/http://goback/ -
8/13/2019 Presentation Qhjt 1final
8/24
-
8/13/2019 Presentation Qhjt 1final
9/24
Quantum Hamilton-Jacobi EquationMarco Roncadelli, L.S. Schulman, Phys. Rev. Lett. 99, 170406 (2007)
Roncadelli and Schulman proved that we can find solutions to the operatorQuantum Hamilton-Jacobi Equation by a simple prescription from thesolutions of the Schrdinger equation for the same Hamiltonian. To provethis, we use a fairly general Weyl-ordered Hamiltonian :
H(q, p, t) =1
2aij(q) pipj+ piaij(q) pj+
1
2pipjaij(q)+bi(q) pi+ pibi(q)+c(q)
where aij(), bi(), and c() are functions ofqk.
Quantum Hamilton Jacobi Theory April 15, 2013 9 / 24
http://find/http://goback/ -
8/13/2019 Presentation Qhjt 1final
10/24
Quantum Hamilton-Jacobi EquationMarco Roncadelli, L.S. Schulman, Phys. Rev. Lett. 99, 170406 (2007)
Using pi= Wqi
, the Quantum Hamilton-Jacobi Equation is :
12aij(q)W
qiWqj
+ Wqi
aij(q)Wqj
+ 12Wqi
Wqj
aij(q)
+bi(q)W
qi+W
qibi(q) +c(q) +
W
t =0
Quantum Hamilton Jacobi Theory April 15, 2013 10 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
11/24
Quantum Hamilton-Jacobi EquationMarco Roncadelli, L.S. Schulman, Phys. Rev. Lett. 99, 170406 (2007)
We sandwich this equation between the eigenstates q| and |Q, and aftera long derivation, we arrive at the c-number Quantum Hamilton Jacobiequation
2aij(q)W(q,Q, t)
qiW(q,Q, t)
qji
2
W(q,Q, t)qiqj
+
2
bi(q)i
aij(q)
qj
W(q,Q, t)
qi+c(q)i
bi(q)
qi
2
22aij(q)qiqj
+ W(q,Q, t)t
=0
Quantum Hamilton Jacobi Theory April 15, 2013 11 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
12/24
Quantum Hamilton-Jacobi EquationMarco Roncadelli, L.S. Schulman, Phys. Rev. Lett. 99, 170406 (2007)
But if we set
(q,Q, t)exp{iW(q,Q, t)}
in the Schrdinger equation, we find the exact same c-number QuantumHamilton Jacobi equation!
Quantum Hamilton Jacobi Theory April 15, 2013 12 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
13/24
Quantum Hamilton-Jacobi EquationMarco Roncadelli, L.S. Schulman, Phys. Rev. Lett. 99, 170406 (2007)
Thus the wavefunction (q,Q, t) gives us the solution, W(q,Q, t), to the
c-number Quantum Hamilton Jacobi equation. From this solution we candirectly find the solution, W(q,Q, t) to the Quantum Hamilton-Jacobiequation by replacing variables by operators, since well-ordering eliminatesany possibility of ambiguity.
Quantum Hamilton Jacobi Theory April 15, 2013 13 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
14/24
Quantum Hamilton-Jacobi EquationMarco Roncadelli, L.S. Schulman, Phys. Rev. Lett. 99, 170406 (2007)
We can show that the wavefunction (q,Q, t) is just the quantumpropagator K(q,Q, t). Since both functions obey the same equation, all
we need to show is that they have the same boundary conditions at t=0.
For the propagator K(q,Q, t) =(qQ) at t=0.
Then we must focus on (q,Q, t) or W.
Quantum Hamilton Jacobi Theory April 15, 2013 14 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
15/24
Quantum Hamilton-Jacobi EquationMarco Roncadelli, L.S. Schulman, Phys. Rev. Lett. 99, 170406 (2007)
We assume that for a nonsingular potential, the solution W approachesthat of the free particle for t0. Thus, W(q,Q) = m2t(Qq)
2.To conserve a well-ordered operator, we must have :
W =m
2t(Q2
2qQ+ q2) +g(t)
With the Quantum Hamilton Jacobi Equation, we have :
W
t
=m
2t2(Q2 2qQ+ q2) +
g
t
=m
2t2(Q2 qQ Qq+ q2)
0= m
2t2[q,Q] +
g(t)
t
Quantum Hamilton Jacobi Theory April 15, 2013 15 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
16/24
Quantum Hamilton-Jacobi EquationMarco Roncadelli, L.S. Schulman, Phys. Rev. Lett. 99, 170406 (2007)
For small t, we can compute the commutator thanks the the relation :
q= Q+Pt
m
Thus, we find :g(t)t
= 12t
(QPPQ) = i
2t
W =m
2t
(Q2 2qQ+ q2) + i
2
ln(t) +const
(q,Q, t) =const
1
texp
i
m
2t(Q2 2qQ+q2)
Quantum Hamilton Jacobi Theory April 15, 2013 16 / 24
Q H l J b E
http://find/ -
8/13/2019 Presentation Qhjt 1final
17/24
Quantum Hamilton-Jacobi EquationMarco Roncadelli, L.S. Schulman, Phys. Rev. Lett. 99, 170406 (2007)
Since we can find any solution of the Schdinger equation by convolving anarbitrary wave function with the propagator, it follows that any solution of
the operator QHJE can ultimately be constructed in terms of the propagator.
(q) =e iW(q)
Quantum Hamilton Jacobi Theory April 15, 2013 17 / 24
Q H il J bi E i E l
http://find/ -
8/13/2019 Presentation Qhjt 1final
18/24
Quantum Hamilton-Jacobi Equation : ExampleSimple Harmonic Oscillator
H= p2
2m+
12m2q2
aij=ij1
4bi=0
c=1
2m2q2
The Quantum Hamilton-Jacobi Equation is :
1
2m
W
q
2+
1
2m2q2 +
W
t =0
Quantum Hamilton Jacobi Theory April 15, 2013 18 / 24
Q H il J bi E i E l
http://goforward/http://find/http://goback/ -
8/13/2019 Presentation Qhjt 1final
19/24
Quantum Hamilton-Jacobi Equation : ExampleSimple Harmonic Oscillator
Converting to c-number form :
1
2m W
q 2
i
2m
2W
q2 +
1
2m2q2 +
W
t =0
We have the propagator :
K(Q, q, t) =
m
2isin(t)exp
im((Q2 +q2) cos(t)2qQ)
2sin(t)
which is a known solution in Quantum Mechanics
Quantum Hamilton Jacobi Theory April 15, 2013 19 / 24
Q t H ilt J bi E ti E l
http://goforward/http://find/http://goback/ -
8/13/2019 Presentation Qhjt 1final
20/24
Quantum Hamilton-Jacobi Equation : ExampleSimple Harmonic Oscillator
Since K(q,Q, t) =expiW(q,Q, t)
, we can easily find the solution :
W(q,Q, t) =m((q2 +Q2) cos(t)2qQ)
2sin(t) +
i
2 lnsin(t)
The quantum solution is then :
W(q, Q, t) =m((q2 +Q2) cos(t)2qQ)
2sin(t)
+ i
2
lnsin(t)
Quantum Hamilton Jacobi Theory April 15, 2013 20 / 24
Q t H ilt J bi E ti E l
http://find/ -
8/13/2019 Presentation Qhjt 1final
21/24
Quantum Hamilton-Jacobi Equation : ExampleSimple Harmonic Oscillator
We can find the time-dependence of the operators :
p=W
q =mqcos(t) Qm
sin(t)
P=W
Q=mQcos(t) + qm
sin(t)
Quantum Hamilton Jacobi Theory April 15, 2013 21 / 24
Quantum Hamilton Jacobi Equation : Example
http://goforward/http://find/http://goback/ -
8/13/2019 Presentation Qhjt 1final
22/24
Quantum Hamilton-Jacobi Equation : ExampleSimple Harmonic Oscillator
Which then give us the solution to the Quantum harmonic oscillator :
q(t) = Pm
sin(t) + Qcos(t)
p(t) =Pcos(t) +mQsin(t)
Quantum Hamilton Jacobi Theory April 15, 2013 22 / 24
http://find/ -
8/13/2019 Presentation Qhjt 1final
23/24
Quantum Hamilton Jacobi Equation
-
8/13/2019 Presentation Qhjt 1final
24/24
Quantum Hamilton-Jacobi EquationConclusion
We can construct solutions to the operator QHJE using the quantumpropagator K(q,Q,t) for the same Hamiltonian.
Once K(q,Q,t) is known, we may get its complex phase W(q,Q,t) andby demanding well-ordering, we produce the operator solutionW(Q, q, t).
Conversely, we can also solve the classical partial differential equation
for W(q,Q,t) to find the propagator K(q,Q,t).
Quantum Hamilton Jacobi Theory April 15, 2013 24 / 24
http://find/