presentation qhjt 1final

8/13/2019 Presentation Qhjt 1final

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Quantum Hamilton Jacobi Theory

April 15, 2013

Quantum Hamilton Jacobi Theory April 15, 2013 1 / 24
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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.

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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)

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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)

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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)

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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)

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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).



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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.

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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

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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

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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!

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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.

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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.

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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

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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)


Q H l J b E
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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)


Q H il J bi E i E l
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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


Q H il J bi E i E l


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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


Q t H ilt J bi E ti E l


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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)


Q t H ilt J bi E ti E l
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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 Equation : Example


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Which then give us the solution to the Quantum harmonic oscillator :

q(t) = Pm

sin(t) + Qcos(t)

p(t) =Pcos(t) +mQsin(t)

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Quantum Hamilton Jacobi Equation


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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).

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