advanced quantum mechanics 2 lecture 1 quantum … · quantum mechanics and classical physics...

Quantum Mechanics and classical physicsQuantisation schemes of quantum mechanics

Advanced Quantum Mechanics 2lecture 1

Quantum Mechanics and classical physics

Yazid Delenda

Departement des Sciences de la matiereFaculte des Sciences - UHLB

http://theorique05.wordpress.com/f411/

Batna, 11 October 2014

1/49 Advanced Quantum Mechanics 2 - lecture 1

Formalisms of classical mechanics

Formalisms of classical mechanicsThe objects that classical physics deals with are material pointswith generalised coordinates q = {qi} = {q1, q2, · · · , qn}, andgeneralised momenta p = {pi} = {p1, p2, · · · , pn}, where we have:

p.q =∑i

There are two formalisms in classical physics: Lagrangian andHamiltonian.In the Lagrangian formalism the laws of motion arethen deduced by the principle of extremum of the action S:

∫ tf

L({qi}, {qi}, t)dt = extremum

where L = T − V , with T the kinetic energy and V the potentialenergy of the system.This results in the Euler-Lagrange equationsof motions:

∂qi− d

∂qi= 0

which are n equations. 2/49 Advanced Quantum Mechanics 2 - lecture 1

p.q =∑i

∫ tf

∂qi− d

∂qi= 0

p.q =∑i

∫ tf

∂qi− d

∂qi= 0

p.q =∑i

∫ tf

∂qi− d

∂qi= 0

p.q =∑i

∫ tf

∂qi− d

∂qi= 0

p.q =∑i

∫ tf

∂qi− d

∂qi= 0

In the Hamiltonian formalism we define the conjugate momenta:

pi =∂L

and the Hamiltonian is:

H =∑

piqi − L

From the Euler lagrange equations one finds that the hamiltonianis a conserved quantity, dH/dt = 0.Then one finds the Hamiltonequations of motion:

q =∂H∂p

, p = −∂H∂q

pi =∂L

H =∑

piqi − L

q =∂H∂p

, p = −∂H∂q

pi =∂L

H =∑

piqi − L

q =∂H∂p

, p = −∂H∂q

pi =∂L

H =∑

piqi − L

q =∂H∂p

, p = −∂H∂q

Problems with classical physics

Classical Physics fails to explain the following phenomenon:

Ultraviolet catastrophe (black body radiation): Before thequantum theory the Raleigh-Jeans law for black bodyradiation predicted that the average power per unit areadiverges in the ultraviolet regime.

Absence of magnetic monopoles: the classical theory is unableto explain the Bohr Magneton.

Atomic spectral lines and stability of atoms: Discreet emissionand absorption lines from atoms is a concept completelyunexplained by classical physics. Furthermore the classicaltheory of electrodynamics predicts that electrons orbitingnuclei should emit synchrotron radiation and lose energy andspiral down to the nucleus.

Formalisms of quantum mechanics

The objects that quantum physics deals with are complex vectorsin Hilbert space |ψ〉 (their components are in general complexnumbers). The dynamical laws follow from the Schrodingerequation:

∂t|ψ〉 = H|ψ〉

Measurements in quantum mechanics lead to the collapse of thestate. Given that the state of the system |ψ〉 is written in a basisof the eigenstates of the measured observable:

|ψ〉 =∑

an|φn〉

then upon measurement of the observable the probability of findingthe state in the state |φn〉 is |an|2, and the state of the systemcollapses into the state |φn〉 after this measurement.

∂t|ψ〉 = H|ψ〉

|ψ〉 =∑

an|φn〉

∂t|ψ〉 = H|ψ〉

|ψ〉 =∑

an|φn〉

∂t|ψ〉 = H|ψ〉

|ψ〉 =∑

an|φn〉

∂t|ψ〉 = H|ψ〉

|ψ〉 =∑

an|φn〉

Thus quantum mechanics is not a generalisation of classicalmechanics.The principle of quantum mechanics is the quantisationscheme while in classical mechanics the principle is the extremumof the action.In physics usually the connection between an old theory and a new(improved) theory is understood through the relation “old ⊂new”.But the connection between quantum and classicalmechanics is not as simple.Classical physics is in some sense a limiting case of quantummechanics (setting ~→ 0), but classical physics still uses severalideas of quantum mechanics to explain collective behaviour ofclassical objects(because macroscopic systems are after all made ofquantum objects).

On the other hand in order to be able to make measurements inquantum mechanics (the concept of “collapse” of wavefunction)classical physics is very much relevant (because our way ofmeasuring the universe is classical).The strangeness with quantum mechanics is that we canunderstand well the equations that govern its rules and makepredictions which fit experiments very well, however when it comesto connecting what we calculate and measure to our commonsense we completely fail.In this regard Richard Feynmann said:“It is impossible to understand quantum mechanics, in the endpeople just get used to it.”

Paradoxes of quantum mechanicsWave-particle duality

Consider the classic 2-slit experiment in which a stream ofelectrons are split into two beams passing through the slits anddetected at a screen.

Screen

The pattern observed in the screen represents the probability offinding the electrons in a given position in the screen.Closing one of the slits changes this probability and thus changesthe observed pattern.The probability of finding an electron in agiven position in the screen in the double slit experiment is not thesum of the probabilities from the two slits separately, i.e. there issome kind of interference exactly of the same nature as wavesinterfering with each other.Thus electrons (which are particles)may be ascribed by some wavefunction having wave properties.

Paradoxes of quantum mechanicsSchrodinger cat:

Consider a two-level atomic system with a wave-function:

ψ(t) = ψ2 exp(−λt) +√

1− exp(−2λt)ψ1

with ψ1 and ψ2 the eigenfunctions of the hamiltonian for thisatomic system.At time t = 0, the state of the system is ψ2 (theexcited state). At later times the system is in a superposition ofthe two states so we do not know whether it is in the excited stateor ground state.The atom can make random transitions to theground state and emit a photon.This photon is then picked up by aphoto-multiplier and a signal is generated triggering a laser aimeda cat.

ψ(t) = ψ2 exp(−λt) +√

If the whole experiment is put in a sealed box then the state of thecat is also a superposition of the form

|ψ〉 = exp(−λt)|alive〉+√

1− exp(−2λt)|dead〉

and we do not know its state prior to any measurement.11/49 Advanced Quantum Mechanics 2 - lecture 1

By making the measurement (looking at the cat) the wavefunctioncollapses into one of the states |alive〉 or |dead〉. But before thatthe state of the cat is a superposition of the two states.Measurement of the state of the cat will only reveal one of thestates, i.e. it is either dead or alive. The actual state of the cat isonly revealed by measurementand we can only say before makingthe measurement that the probability of finding the cat alive isexp(−2λt). The collapse of the wavefunction only occurs uponmaking the measurement. The paradox lies in the fact that theunobserved universe is in a state of “uncollapsedness”.

Paradoxes of quantum mechanicsThe Enstein-Podolsky-Rosen (EPR) Paradox:

Consider the atomic transitions in which two spin-1 photons areemitted in opposite directions. By conservation of angularmomentum the two photons must have opposite spins, if one ofthem has ms = +1 the other must have ms = −1. Measuring thespin state of one photon reveals the spin state of the other. Nowthe paradox lies in the fact that the photons fly away at the speedof light. If the spin state of one of the photons is measured at oneplace in the universe then the spin state of the other immediatelycollapses as measured at that particular point in space. Theproblem with the instantaneous collapse of the state of the otherphoton is in violation with the laws of special relativity.

Orthodox quantisationModern version of Quantum mechanics

Orthodox quantisation

Orthodox quantum mechanics (also known as Copenhagenquantisation) deals with normalised wavefunctionsψ(x) = ψ(x1, x2, · · · , t) which satisfy the normalisation condition:∫

Vψ∗(x1, x2, · · · , t)ψ(x1, x2, · · · , t)dx1dx2... = 1

The laws of motion follow from the Schrodinger equation:

i~∂ψ(x, t)

∂t= Hψ(x, t)

where the Hamiltonian is:

H = − ~2

∂x2+ V (x)

with − ~22m

∂x2the kinetic energy term and V (x) the potential

energy.14/49 Advanced Quantum Mechanics 2 - lecture 1

Vψ∗(x1, x2, · · · , t)ψ(x1, x2, · · · , t)dx1dx2... = 1

i~∂ψ(x, t)

∂t= Hψ(x, t)

H = − ~2

∂x2+ V (x)

with − ~22m

Vψ∗(x1, x2, · · · , t)ψ(x1, x2, · · · , t)dx1dx2... = 1

i~∂ψ(x, t)

∂t= Hψ(x, t)

H = − ~2

∂x2+ V (x)

with − ~22m

Vψ∗(x1, x2, · · · , t)ψ(x1, x2, · · · , t)dx1dx2... = 1

i~∂ψ(x, t)

∂t= Hψ(x, t)

H = − ~2

∂x2+ V (x)

with − ~22m

Measurement procedure for a system described by a wavefunctionψ(x, t) =

∑n anφn(x, t) (where φn are the eigenfunctions of the

measured observable Q with the corresponding eigenvalues qn),resulting in a give eigenvalue qm, leads to the collapse of thewavefunction to φm(x, t) with probability |am|2.The correspondence rule is the first quantification, i.e. weassociate with each measured quantity an operator. So if theclassical Hamiltonian is a function of coordinates H(p, q) then wepromote the position and momentum into operators:

q → x, p→ −i~ ∂∂x, H → H = i~

Paradoxes with this scheme may be written as:

Ordering of operators in classical physics is irrelevant, sof(p, q) = p2q = pqp = qp2. However in quantum mechanics itis important, so one gets various forms of the operator f byintroducing the quantisation (3 forms in the above example).Changing variables in classical mechanics is a liberty, forinstance one can introduce p′ =

√p and q′ = q4, in which

case in the above example we can write f(p, q) = p′4 4√q′. But

in quantum mechanics this change of variable leads todifferent forms of operators and thus different eigenstates.The normalisation of the wavefunction:∫

Vψ∗(x)ψ(x)dx = 1

means the number of particles is conserved, so no particlecreation/distruction is allowed in this quantisation scheme,

which is a problem when dealing with situations in which particlecreation/annihilation is allowed.

Constraint of eigenvalues of operators in quantum mechanics(thus on values of measured physical quantities), while inclassical physics there are no constraints on values of physicalquantities.

To see the last point let us consider for example a particle of massm moving in a one-dimensional space in a potentialV (x) = u0δ(x), with δ(x) the δ-Dirac function. So the TISESchrodinger equation is:

− ~2

∂2ψ(x)

∂x2+ V (x)ψ(x) = Eψ(x)

with E < 0 for bound states. Here ψ the eigenfunction and E theeigenvalue of the Hamiltonian. Also V (x) = u0δ(x).

− ~2

∂2ψ(x)

∂x2+ V (x)ψ(x) = Eψ(x)

− ~2

∂2ψ(x)

∂x2+ V (x)ψ(x) = Eψ(x)

− ~2

∂2ψ(x)

∂x2+ V (x)ψ(x) = Eψ(x)

− ~2

∂2ψ(x)

∂x2+ V (x)ψ(x) = Eψ(x)

− ~2

∂2ψ(x)

∂x2+ V (x)ψ(x) = Eψ(x)

− ~2

∂2ψ(x)

∂x2+ V (x)ψ(x) = Eψ(x)

For x 6= 0 we may simplify the TISE as:

− ~2

∂2ψ(x)

∂x2= Eψ(x)

where δ(x) = 0 for x 6= 0. The solution to this equation isrelatively straightforward:

ψ(x) = Ae−kx +Be+kx

where k > 0, and substituting into the TISE we obtain:

√−2mE

To ensure that the wave-function be finite at ±∞ we may thuswrite:

ψ(x) = AΘ(x)e−kx +BΘ(−x)e+kx

− ~2

∂2ψ(x)

∂x2= Eψ(x)

√−2mE

− ~2

∂2ψ(x)

∂x2= Eψ(x)

√−2mE

− ~2

∂2ψ(x)

∂x2= Eψ(x)

√−2mE

− ~2

∂2ψ(x)

∂x2= Eψ(x)

√−2mE

with Θ(x) the heaviside theta function:

Θ(x) = + 1, x > 0

Θ(x) =0, x < 0

Back to the Schrodinger equation, we can write:

− ~2

∂2ψ(x)

∂x2+ u0δ(x)ψ(x) = Eψ(x)

Integrating both sides from −ε to +ε, with ε→ 0 (i.e. a very smallvariable which we set to zero at the end of the calculation) weobtain:

− ~2

∫ +ε

∂2ψ(x)

∂x2dx+ u0

∫ +ε

−εδ(x)ψ(x)dx =E

∫ +ε

−εψ(x)dx

≈E∫ +ε

−εψ(0)dx = 2Eε→ 0

Θ(x) = + 1, x > 0

Θ(x) =0, x < 0

− ~2

∂2ψ(x)

∂x2+ u0δ(x)ψ(x) = Eψ(x)

− ~2

∫ +ε

∂2ψ(x)

∂x2dx+ u0

∫ +ε

−εψ(x)dx

≈E∫ +ε

−εψ(0)dx = 2Eε→ 0

Θ(x) = + 1, x > 0

Θ(x) =0, x < 0

− ~2

∂2ψ(x)

∂x2+ u0δ(x)ψ(x) = Eψ(x)

− ~2

∫ +ε

∂2ψ(x)

∂x2dx+ u0

∫ +ε

−εψ(x)dx

≈E∫ +ε

−εψ(0)dx = 2Eε→ 0

Θ(x) = + 1, x > 0

Θ(x) =0, x < 0

− ~2

∂2ψ(x)

∂x2+ u0δ(x)ψ(x) = Eψ(x)

− ~2

∫ +ε

∂2ψ(x)

∂x2dx+ u0

∫ +ε

−εψ(x)dx

≈E∫ +ε

−εψ(0)dx = 2Eε→ 0

Θ(x) = + 1, x > 0

Θ(x) =0, x < 0

− ~2

∂2ψ(x)

∂x2+ u0δ(x)ψ(x) = Eψ(x)

− ~2

∫ +ε

∂2ψ(x)

∂x2dx+ u0

∫ +ε

−εψ(x)dx

≈E∫ +ε

−εψ(0)dx = 2Eε→ 0

which tens to zero as ε→ 0. Further more we have from theproperties of the Dirac function:∫ +ε

−εδ(x)ψ(x) = ψ(0)dx

− ~2

∫ +ε

∂2ψ(x)

∂x2dx+ u0ψ(0) =0 (1)

The remaining integral is easy since the integrand is a totalderivative:∫ +ε

∂2ψ(x)

∂x2dx =

∫ +ε

(dψ(x)

dψ(x)

∣∣∣∣+ε−ε

− ~2

∫ +ε

∂2ψ(x)

∂x2dx+ u0ψ(0) =0 (1)

∂2ψ(x)

∂x2dx =

∫ +ε

(dψ(x)

dψ(x)

∣∣∣∣+ε−ε

− ~2

∫ +ε

∂2ψ(x)

∂x2dx+ u0ψ(0) =0 (1)

∂2ψ(x)

∂x2dx =

∫ +ε

(dψ(x)

dψ(x)

∣∣∣∣+ε−ε

− ~2

∫ +ε

∂2ψ(x)

∂x2dx+ u0ψ(0) =0 (1)

∂2ψ(x)

∂x2dx =

∫ +ε

(dψ(x)

dψ(x)

∣∣∣∣+ε−ε

then using the form of the wave function at x > 0, ψ(x) ∼ e−kxwe obtain (by setting ε→ 0)

dψ(x)

∣∣∣∣+ε = −kψ(0)

and similarly using the form at x < 0, ψ(x) ∼ e+kx, we obtain(setting ε→ 0)

dψ(x)

∣∣∣∣−ε

= +kψ(0)

then substituting into (1):

− ~2

2m[−2kψ(0)] + u0ψ(0) = 0

dψ(x)

∣∣∣∣+ε = −kψ(0)

dψ(x)

∣∣∣∣−ε

= +kψ(0)

− ~2

2m[−2kψ(0)] + u0ψ(0) = 0

dψ(x)

∣∣∣∣+ε = −kψ(0)

dψ(x)

∣∣∣∣−ε

= +kψ(0)

− ~2

2m[−2kψ(0)] + u0ψ(0) = 0

dψ(x)

∣∣∣∣+ε = −kψ(0)

dψ(x)

∣∣∣∣−ε

= +kψ(0)

− ~2

2m[−2kψ(0)] + u0ψ(0) = 0

dψ(x)

∣∣∣∣+ε = −kψ(0)

dψ(x)

∣∣∣∣−ε

= +kψ(0)

− ~2

2m[−2kψ(0)] + u0ψ(0) = 0

from which it follows that

k = −u0m~2

which constrains the energy (Note this also means that u0 must benegative to have bound states):

E = −~2k2

2m= −u

Note that from the boundary conditions (continuity of thewavefunction) we must have A = B, and to ensure normalisationof the wavefunction we must have A =

√k, so the wave function

may be written as:

ψ(x) =√k(

Θ(x)e−kx + Θ(−x)e+kx)

k = −u0m~2

E = −~2k2

2m= −u

may be written as:

ψ(x) =√k(

k = −u0m~2

E = −~2k2

2m= −u

may be written as:

ψ(x) =√k(

k = −u0m~2

E = −~2k2

2m= −u

may be written as:

ψ(x) =√k(

k = −u0m~2

E = −~2k2

2m= −u

may be written as:

ψ(x) =√k(

k = −u0m~2

E = −~2k2

2m= −u

may be written as:

ψ(x) =√k(

k = −u0m~2

E = −~2k2

2m= −u

may be written as:

ψ(x) =√k(

Modern version of Quantum mechanics

The modern version of quantum mechanics is described by Diracformalism, in which objects are state vectors belonging to statespace, also known as kets , and the dynamics law is once again theSchrodinger equation:

H|ψ〉 = i~∂

∂t|ψ〉

and the principle of quantum mechanics is also the quantisationscheme (first quantisation)

H|ψ〉 = i~∂

∂t|ψ〉

H|ψ〉 = i~∂

∂t|ψ〉

Linearity

Starting from the Schrodinger equation for a wavefunction ψ(x, t)with time-independent Hamiltonian:

∂tψ(x, t) = Hψ(x, t) =

(− ~2

∂x2+ V (x)

)ψ(x, t)

and assuming that the initial state ψi = ψ(x, 0) of the system isknown. We may write by taking the Schrodinger equation at timet = 0:

i~∂ψ(x, t)

∣∣∣∣t=0

= Hψ(x, 0)

where by the definition of the derivative of a function we have:

i~ψ(x, δt)− ψ(x, 0)

δt= Hψ(x, 0)⇒ ψ(x, δt)−ψ(x, 0) = − i

~Hψ(x, 0)δt

with δt→ 0.25/49 Advanced Quantum Mechanics 2 - lecture 1

Linearity

∂tψ(x, t) = Hψ(x, t) =

(− ~2

∂x2+ V (x)

)ψ(x, t)

i~∂ψ(x, t)

∣∣∣∣t=0

= Hψ(x, 0)

i~ψ(x, δt)− ψ(x, 0)

δt= Hψ(x, 0)⇒ ψ(x, δt)−ψ(x, 0) = − i

~Hψ(x, 0)δt

Linearity

∂tψ(x, t) = Hψ(x, t) =

(− ~2

∂x2+ V (x)

)ψ(x, t)

i~∂ψ(x, t)

∣∣∣∣t=0

= Hψ(x, 0)

i~ψ(x, δt)− ψ(x, 0)

δt= Hψ(x, 0)⇒ ψ(x, δt)−ψ(x, 0) = − i

~Hψ(x, 0)δt

Linearity

∂tψ(x, t) = Hψ(x, t) =

(− ~2

∂x2+ V (x)

)ψ(x, t)

i~∂ψ(x, t)

∣∣∣∣t=0

= Hψ(x, 0)

i~ψ(x, δt)− ψ(x, 0)

δt= Hψ(x, 0)⇒ ψ(x, δt)−ψ(x, 0) = − i

~Hψ(x, 0)δt

Linearity

∂tψ(x, t) = Hψ(x, t) =

(− ~2

∂x2+ V (x)

)ψ(x, t)

i~∂ψ(x, t)

∣∣∣∣t=0

= Hψ(x, 0)

i~ψ(x, δt)− ψ(x, 0)

δt= Hψ(x, 0)⇒ ψ(x, δt)−ψ(x, 0) = − i

~Hψ(x, 0)δt

Linearity

So we have at first order:

ψ(x, δt) =

(1− i

)ψ(x, 0)

Iterating this procedure (i.e. setting now the initial time at δt) wemay write:

ψ(x, 2δt) =

(1− i

)ψ(x, δt)

(1− i

)(1− i

)ψ(x, 0)

ψ(x, nδt) =

(1− i

)nψ(x, 0)

Linearity

So we have at first order:

ψ(x, δt) =

(1− i

)ψ(x, 0)

Iterating this procedure (i.e. setting now the initial time at δt) wemay write:

ψ(x, 2δt) =

(1− i

)ψ(x, δt)

(1− i

)(1− i

)ψ(x, 0)

ψ(x, nδt) =

(1− i

)nψ(x, 0)

Linearity

Defining t ≡ nδt (i.e. we divide the interval [0, t] into n equallyspaced time intervals of length δt), so when δt→ 0 then n→∞.Hence we may write:

ψ(x, t) = limn→∞

(1− i

)nψ(x, 0)

but we know that

limn→∞

(1− x

)n= e−x

so we arrive at:

ψ(x, t) = exp

(− it~H

)ψ(x, 0) = U(t)ψ(x, 0)

U(t) = exp

(− it~H

)27/49 Advanced Quantum Mechanics 2 - lecture 1

Linearity

(1− i

)nψ(x, 0)

but we know that

limn→∞

(1− x

)n= e−x

so we arrive at:

ψ(x, t) = exp

(− it~H

)ψ(x, 0) = U(t)ψ(x, 0)

U(t) = exp

(− it~H

Linearity

(1− i

)nψ(x, 0)

but we know that

limn→∞

(1− x

)n= e−x

so we arrive at:

ψ(x, t) = exp

(− it~H

)ψ(x, 0) = U(t)ψ(x, 0)

U(t) = exp

(− it~H

Linearity

(1− i

)nψ(x, 0)

but we know that

limn→∞

(1− x

)n= e−x

so we arrive at:

ψ(x, t) = exp

(− it~H

)ψ(x, 0) = U(t)ψ(x, 0)

U(t) = exp

(− it~H

Linearity

(1− i

)nψ(x, 0)

but we know that

limn→∞

(1− x

)n= e−x

so we arrive at:

ψ(x, t) = exp

(− it~H

)ψ(x, 0) = U(t)ψ(x, 0)

U(t) = exp

(− it~H

Linearity

(1− i

)nψ(x, 0)

but we know that

limn→∞

(1− x

)n= e−x

so we arrive at:

ψ(x, t) = exp

(− it~H

)ψ(x, 0) = U(t)ψ(x, 0)

U(t) = exp

(− it~H

Linearity

and if we set the initial time at t0 then:

ψ(x, t) = exp

(− i(t− t0)

)ψ(x, 0) = U(t, t0)ψ(x, 0)

U(t, t0) = exp

(− i(t− t0)

)This operator is called the evolution operator and it is unitary,meaning that:

U †U = 1

which follows from the hermitian Hamiltonian H† = H.

Linearity

ψ(x, t) = exp

(− i(t− t0)

)ψ(x, 0) = U(t, t0)ψ(x, 0)

U(t, t0) = exp

(− i(t− t0)

U †U = 1

Linearity

ψ(x, t) = exp

(− i(t− t0)

)ψ(x, 0) = U(t, t0)ψ(x, 0)

U(t, t0) = exp

(− i(t− t0)

U †U = 1

Linearity

ψ(x, t) = exp

(− i(t− t0)

)ψ(x, 0) = U(t, t0)ψ(x, 0)

U(t, t0) = exp

(− i(t− t0)

U †U = 1

Linearity

ψ(x, t) = exp

(− i(t− t0)

)ψ(x, 0) = U(t, t0)ψ(x, 0)

U(t, t0) = exp

(− i(t− t0)

U †U = 1

Linearity

In the case that the hamiltonian is time-dependent the evolutionoperator is not so simple:

U(t, t0) = T

(−i~

H(τ)dτ

)}where T is the time-ordering (Wick’s normal ordering) operator[More on this in Relativistic quantum mechanics course].

Linearity

In the case that the hamiltonian is time-dependent the evolutionoperator is not so simple:

U(t, t0) = T

(−i~

H(τ)dτ

)}where T is the time-ordering (Wick’s normal ordering) operator[More on this in Relativistic quantum mechanics course].

Linearity

Now for a linear combination of two wave functions at initial timeti, ψ1(x, ti) and ψ2(x, ti), by ψ(x, ti) = αψ1(x, ti) + βψ2(x, ti),the evolution of the superposed wavefunction is actually thesuperposition of the evolution of the two wavefunctions. So at alater time we have: