Advanced Quantum Mechanics

Home , Time evolution

Rajdeep Sensarma

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Quantum Dynamics

Lecture #9 Schrodinger and Heisenberg Picture Time Independent Hamiltonian

Schrodinger Picture: A time evolving state in the Hilbert space with time independent operators

iHtˆ i@t (t) = Hˆ (t) (t) = e (0) | i | i | i | i

i✏nt Eigenbasis of Hamiltonian: Hˆ n = ✏ n (t) = cn(t) n c (t)=c (0)e n | i | i n n | i | i n X

iHtˆ iHtˆ Operators and Expectation: A(t)= (t) Aˆ (t) = (0) e Aeˆ (0) = (0) Aˆ(t) (0) h | | i h | | i h | | i

Heisenberg Picture: A static initial state and time dependent operators

iHtˆ iHtˆ Aˆ(t)=e Aeˆ

Schrodinger Picture Heisenberg Picture

(0) (t) (0) (0) | i!| i | i!| i Equivalent description iHtˆ iHtˆ Aˆ Aˆ Aˆ Aˆ(t)=e Aeˆ ! ! of a quantum system i@ (t) = Hˆ (t) i@ Aˆ(t)=[Aˆ(t), Hˆ ] t| i | i t Time Evolution and Propagator

iHtˆ (t) = e (0) Time Evolution Operator | i | i

(x, t)= x (t) = x Uˆ(t) (0) = dx0 x Uˆ(t) x0 x0 (0) h | i h | | i h | | ih | i Z

Propagator:

Example : Free Particle

The propagator satisﬁes

and hence is often called the Green’s function Retarded and Advanced Propagator

The following propagators are useful in different contexts

Retarded or Causal Propagator:

This propagates states forward in time for t > t’

Advanced or Anti-Causal Propagator:

This propagates states backward in time for t < t’

Both the retarded and the advanced propagator satisﬁes the same diff. eqn., but with different boundary conditions Propagators in Frequency Space

Energy Eigenbasis |n>

The integral is ill deﬁned due to oscillatory nature of the integrand Integrals for retarded/advanced propagators can be made well deﬁned in the following way:

Retarded Propagator:

This integral is convergent if we replace ω —> ω+i η, with η—> 0+

Positive η provides exponential decay of integrand at large +ve t Propagators in Frequency Space

Advanced Propagator:

This integral is convergent if we replace ω —> ω- i η, with η—> 0+

Negative η provides exponential decay of integrand at large -ve t

Note that the original integral from -∞ to ∞ cannot be made well deﬁned by either prescription

Example: Free Particle

Example: Harmonic Oscillator Propagators in Fourier Space Let us make sure that the prescription of adding (subtracting) a small positive imaginary part to the frequency gives back the correct retarded (advanced) propagator on inverse transformation

Retarded Propagator:

• Evaluate the ω integral by contour integration. With ω—> ω+ i η, the poles of the integrand are all in lower half-plane.

• For t>0, use the blue contour to do the integration. Note that for t > 0, the integral on the semicircle vanishes. ω

⤫ ⤫ ⤫ • For t<0, one cannot use the blue contour. For large -ve imaginaryωthe integrand blows up. So, instead use the red contour. Since there are no singularities in the upper half-plane, we get 0. Propagators in Fourier Space Let us make sure that the prescription of adding (subtracting) a small positive imaginary part to the frequency gives back the correct retarded (advanced) propagator on inverse transformation

Advanced Propagator:

• Evaluate the ω integral by contour integration. With ω—> ω- i η, the poles of the integrand are all in upper half-plane.

• For t>0, use the blue contour to do the integration. Since all the singularities are in lower half-plane, we get 0 ω • For t<0, one cannot use the blue contour. For large -ve ⤫ ⤫ ⤫ imaginaryωthe integrand blows up. So, instead use the red contour. Pick up residues from the poles to get

So in all Time Dependent Hamiltonians

Where do we encounter time dependent Hamiltonians?

✦Experimental probes of systems: ๏ When we shine light on atoms and measure absorption, we turn on the part of H corresponding to light-matter interaction and switch it off. The absorption is described by a time dependent Hamiltonian ๏ When we apply voltage and measure current, we are measuring the response of the system to a time dependent Hamiltonian (part corresponding to energy of charges in E ﬁeld) ๏ In NMR experiments, we turn on rf pulses, which correspond to a time dependent Hamiltonian

✦Tuning Hamiltonian (Interaction) parameters: In cold atomic systems, the effective interaction between particles can be tuned by changing a magnetic ﬁeld in a time dependent way. The main difference between this and above is that we do not necessarily turn off the interaction

✦ Coupling to a Bath: If we wish to understand how quantum systems equilibriate, we have to study the problem of a system coupled to a heat bath. Under certain conditions, this problem reduces to a quantum system acted on by a time-dependent random noise. E.g.: Quantum Brownian Particle, Oscillator kicked by noise from you walking around it, etc. Propagator for time dependent Hamiltonians Formal Solution for time-dependent Hamiltonians: Integral Equation

Integrate t i@ (t) = Hˆ (t) (t) (t) = (0) i dt0Hˆ (t0) (t0) t | i | i | i | i | i Z0

. . n t t1 tj 1 (n)(t) = [1+ ( i)j dt dt ... dt Hˆ (t )Hˆ (t )...Hˆ (t )] (0) | i 1 2 j 1 2 j | i j=1 0 0 0 X Z Z Z Note that time argument of H s are decreasing as we go from left to right —————> Time ordering

j t t1 tj 1 (t) =[1+ 1 ( i) dt dt ... dt Hˆ (t )Hˆ (t )...Hˆ (t )] (0) | i j=1 0 1 0 2 0 j 1 2 j | i

t Pi dt0Hˆ (t0) R R R = T e 0 (0) R | h i Time Ordering We have formally written the time evolution operator for a time dependent Hamiltonian as a time- ordered exponential. What does this mean?

In the jth order term above, there are j! possible orderings of {t1,t2,..tj}. (say {t1>t2>..tj}, {t2>t1>t3>..tj} and so on).

The time ordering operator takes any of this j! terms and simply replaces it by the ordering {t1>t2>..tj}. This results in (a) cancellation of the j! in the expression and (b) change in the limits of integration to reﬂect this ordering.

is a correct but almost always useless formula by itself (i.e. without any further approximations).

It is however a good starting point for making various approximations. Time Dependent Hamiltonians

The main reason that time-dependent Hamiltonians are harder to work with is that the Hamiltonian operator at different times do not commute with each other.

Example: A Harmonic oscillator kicked by a spatially uniform time-dependent external force

2 ˆ pˆ 1 2 2 ˆ ˆ i~(f(t1) f(t2)) H(t)= 2m + 2 m!0xˆ + f(t)ˆx [H(t1), H(t2)] = m pˆ

Since Hamiltonians at different times do not commute, we cannot diagonalize them simultaneously. There is thus no obvious choice of basis to work with.

We will look at different approximation schemes to deal with time-dependent Hamiltonians, which involve making different choice of basis to expand the problem • Time Dependent Perturbation Theory: Time dependent part of the Hamiltonian is parametrically small. Expand in eigenbasis of time-independent part of the Hamiltonian.

• Adiabatic Limit: The rate of change of Hamiltonian is parametrically small. Expand in the time dependent basis which diagonalizes the instantaneous Hamiltonian.

• Rotating Wave Approximation: Work with a time-dependent basis and neglect the fast varying part of the Hamiltonian to get an effective time-independent Hamiltonian. Work with eigenbasis of this effective Hamiltonian Interaction Representation

Consider a Hamiltonian formed of two parts Hˆ = Hˆ0 + Hˆ1

The breakup can be motivated by different considerations:

Hˆ = Hˆ0 + Hˆ1 Hˆ = Hˆ0 + Hˆ1 Hˆ = Hˆ0 + Hˆ1

Part that cannot A Hamiltonian Time indep. part Time dep. part Hamiltonian Probe-system be solved e.g. that can be solved of Hamiltonian of Hamiltonian of system Interaction Particle-Particle e.g. free particl e Interaction

Interaction Representation: Put the time evolution due to Hˆ 0 on the operators and the rest on the states.

ˆ Deﬁne (t) = eiH0t (t) | I i | S i

Schrodinger Eqn.: i@ (t) =(Hˆ + Hˆ ) (t) t| S i 0 1 | S i iHˆ t iHˆ t e 0 (Hˆ + i@ ) (t) =(Hˆ + Hˆ )e 0 (t) 0 t | I i 0 1 | I i iHˆ t iHˆ t i@ (t) = e 0 Hˆ e 0 (t) t| I i 1 | I i Interaction Representation

i@ (t) = Hˆ (t) t| I i 1I | I i

Possible time dependence of parameters of H1

iHˆ0t iHˆ0t Deﬁne AˆI (t)=e Aeˆ

for any operator A , i.e. operators evolve as in Heisenberg rep, but with H0

States evolve as in Schrodinger rep., but with the interaction rep. of H1

We will assume that eigenstates and spectrum of Hˆ 0 , or approximations to them, are known

Hˆ n = ✏ n and expand in this basis 0| i n| i

i✏ t (t) = c (t)e n n (t) = c (t) n | S i n n | i)| I i n n | i P P

i(✏n ✏m)t ˆ i@tcn(t)= m e H1mn(t) P Driven Harmonic Oscillator: a special case

Spatially uniform force

iH t iH t i! ta†a i! ta†a Work in the interaction picture : H = e 0 H e 0 = f(t)e 0 (a† + a)e 0 1I 1

Use [a†a, a†]=a† [a†a, a]= a

1 i!0ta†a i!0ta†a n i!0t e a†e = a† + (i!0t) /n![a†a, [a†a, . . . [a†a, a]...]] = a†e n=1 X

i! ta†a i! ta†a i! t Similarly e 0 ae 0 = ae 0

i! t i! t H = f(t)(a†e 0 + ae 0 ) Interaction Picture Hamiltonian: 1I

i! t i! t i! t0 i! t0 [H (t),H (t0)] = f(t)f(t0)[(a†e 0 + ae 0 ), (a†e 0 + ae 0 )] = 2if(t)f(t0)sin[! (t t0)] 1I 1I 0 Constant Driven Harmonic Oscillator: a special case

Crucial in making this [H1I (t), [H1I (t0),H1I (t00)]] = 0 problem a solvable one

Interaction picture i@t I (t) = Hˆ1I I (t) Eqn. of motion: | i | i

t i H (t0)dt0 Time Evolution Operator: U(t, 0) = T [e 0 1I ] R

Total Error ~ N ε2 ———> 0 2ε (N-2)ε t = Nε 0 when N ——>∞, ——> 0, ε 3ε (N-1)ε ε N ✏ 0 N✏ = t so that N is ﬁxed !1 ! ε N 1 m✏ i (m 1)✏ H1I (t0)dt0 U(t, 0) = LimN U(m✏, (m 1)✏)=LimN e !1 !1 m=1 m=1 R Y Y Trotter error due to taking off the time ordering ✏2 ⇠ Driven Harmonic Oscillator

A B A+B+ 1 [A,B] Now e e = e 2 when [A, [A, B]] = 0

(m+1)✏ m✏ (m+1)✏ 1 (m+1)✏ m✏ i H (t0)dt0 i H1I (t0)dt0 dt00 dt0[H1I (t0),H1I (t00)] i m✏ H1I (t00)dt00 (m 1)✏ 1I (m 1)✏ 2 m✏ (m 1)✏ e e = e R R R R R

t 1 t t i H (t0)dt0 dt0 0 dt00[H (t0),H (t00)] U(t, 0) = e 0 1I 2 0 0 1I 1I t i! t0 R R R ⇣(t)= i dt0f(t0)e 0 0 i(t)+⇣(t)a† ⇣⇤(t)a i(t) Z = e = e D[⇣(t)]

t t0 (t)= dt0 dt00f(t0)f(t00)sin[! (t0 t00)] 0 Z0 Z0 Convert back to Schrodinger picture:

iH0t iH0t it i!0t iH0t U(t)=e UI (t)e = e D[⇣(t)e ]e

If we start with the ground state, the drive produces coherent states and upto a phase, the dynamics is that of coherent states.