Download Irreversibility and Quantum Mechanics?

Irreversibility and Quantum Mechanics?
József Verhás
Budapest University of Technology and Economy,
H-1521 Budapest, Hungary∗
[email protected]
http://newton.phy.bme.hu/˜verhas/
Abstract
Basing on Bohr’s correspondence principle and the classical radiation formula, an additional term into Schrödinger’s equation is proposed. The new term makes quantum mechanics irreversible.
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Introduction.
More than three decade ago, in a special course on electrodynamics, optics, and the elements of quantum mechanics, I was asked to calculate the
complex polarizability of a one electron system from quantum mechanics. I
tried hard to do it with the quantum mechanical averages but permanently
failed. I graduated as a chemical engineer but learned—autodidactically—on
mathematics and physics a lot, even concerning quantum theory [1–18]. The
polarizability was given always real by the calculations; Bohr’s correspondence principle was violated. Next I turned to the spontaneous emission,
which is not accounted by Schrödinger’s equation. The hypothesis I started
from was that the transition takes much longer time than the period of the
light quanta and the Fourier coefficients of the expansion of the wave function by the eigenfunctions of the Hamilton operator change slowly, moreover,
the change of the average energy has to be in accordance with the classical
formula of the dipole radiation. The amplitude of the emitted light was
displayed by the equation and agreed perfectly with the result of quantum
electrodynamics.
∗
Supported by OTKA 62278 and T04848
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Time to time I took the problem during the decades but with little success
until having retired. Recently I did some calculations and found a more convincing method to investigate the properties of the solution of Schrödinger’s
equation in the presence of electromagnetic wave.
The results were astonishing for me because I had not learned of the
Rabi oscillation. The functions were hardly be interpreted as transitions; the
Fourier coefficients were oscillating even if the exciting frequency equaled the
proper one perfectly.
It looked like Schrödinger’s electron could not absorb light at all.
A procedure similar to the above, successful for the spontaneous emission,
may give some additional term to Schrödinger’s equation. The new term is
not linear and breaks the symmetry of the equation with respect of time
inversion.
May be that the atomic processes are irreversible?
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The complex polarizability of the electron
is always real.
To prove the statement in the above header assume Schrödinger’s equation.
Hψ = i~
∂ψ
∂t
(2.1)
and assume the transformation T
T : t ←→ −t
i ←→ −i
(2.2)
Here H is the Hamiltonian operator depending on the electric field strength
~ which is supposed a harmonic function of time, E
~ = E
~ 0 cos(ωt). The
E,
boundary condition is assumed homogeneous and the initial condition reads
ψ(0) = eiδ ϕ,
(2.3)
where δ is a phase constant—not depending on anything—and phi is an
eigenfunction of the Hamiltonian operator belonging to the absence of the
~ = 0) if the initial state is a pure one. As the equation 2.1 is
electric field (E
a homogeneous and linear, the solution may be looked for in the form
ψ = eiδ ψ̂.
(2.4)
The function ψ̂ is invariant under transformation (2.2) because both the
differential equation and the initial and boundary conditions are invariant.
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The dipole moment is calculated as
p~ = −e < ψ|~r|ψ >= −e < ψ̂|~r|ψ̂ >,
(2.5)
where −e is the electric charge of the electron. It is obviously invariant under
transformation (2.2); consequently,
p~ = P~0 cos(ωt).
(2.6)
The linear approximation for the polarizability in the usual complex form
reads
~
p~ = χE,
(2.7)
where χ is a real tensor.
If the initial state is mixed the proof starts with the linearization of
Schrödinger’s equation. For the sake of convenience an auxiliary parameter
λ is introduced into the form of the electric field strength;
~ = λE
~ 0 cos(ωt).
E
(2.8)
The power series of the quantities up to the first order read
H = H0 + λH1
ψ = ψ0 + λψ1
(2.9)
and the linearized equation
H0 ψ1 − i~
∂ψ1
= −H1 ψ0 .
∂t
(2.10)
The equation is linear and inhomogeneous; its solution can be obtained term
by term if the right hand side is the sum of a finite or infinite sequence of
expressions. The function ψ0 is combined from the wave functions of pure
states. Their contribution to the complex polarizability tensor has to be
summed up. As the contribution of each is real as has been proved above,
the total polarizability tensor is also real.
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Spontaneous emission.
The common textbooks do not mention the above violation of Bohr’s correspondence principle but confess that the spontaneous emission can not be
accounted on by Schrödinger’s equation. Of course, a term belonging to the
interaction with the emitted electromagnetic field is missing and it is said
that only the quantum theory of it solves the problem. The teachers problem
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is difficult to overcome. One can refer to a theory that is to be taught much
later if ever. As for myself, I did not hope my students would be able to
learn quantum electrodynamics at all. At last I found a simpler way. It is
based on the classical formula of the dipole radiation and the energy balance.
I tested the method on the damped oscillator and fond out that it worked
well if the damping was small. Next I tried to apply the idea at a two level
quantum system.
The solution of Schrödinger’s equation for such a system reads
ψ = c1 ϕ1 e−iω1 t+δ1 + c2 ϕ2 e−iω2 t+δ2 ,
(3.1)
where ϕ1 and ϕ2 are eigenfunctions of the (time independent) Hamiltonian
operator. The coefficients c1 and c2 are real; their angle is shifted to the
exponential functions. With the usual method, the dipole moment is given
as
p~ = −ec1 c2~r12 cos(ωt + δ),
(3.2)
where the notations ω = ω2 − ω1 and δ = δ1 − δ2 have been introduced.
The energy of the system is = c21 ~ω1 + c22 ~ω2 . If c1 and c2 are constants
the energy does not change but the oscillating dipole is emitting energy
according the classical formula
P =
µ0 (p¨~)2
.
6πc
(3.3)
Here the SI units are used; µ0 is the magnetic permeability of the vacuum
and c is the light velocity.
Supposing that the c coefficients do change but slowly we get
2
d
ω4
dc1
dc2
µ0 e2 c21 c22~r12
= 2c1
~ω1 + 2c2
~ω2 =
.
dt
dt
dt
12πc
(3.4)
Making use of the relation c21 + c22 = 1, an equation for the c1 coefficient is
obtained
2
dc1
µ0 e2 c1 (1 − c21 )~r12
ω3
=
.
(3.5)
dt
24π~c
The equation agrees with the results of the quantum theory of the electromagnetic field [12, 17].
The formula is easy to generalize to any mixed states if we assume that
the radiation does not influence the angles of the c coefficients, moreover, the
transitions go on independently.
2
dci X µ0 e2 ci c2k~rik
(ωk − ωi )3
=
.
dt
24π~c
k
4
(3.6)
Here the usual notation ~rik = hϕ1 |~r|ϕk i has been adopted.
To obtain an equation giving account also on the changes of the c coefficients one has to furnish the left hand side of Schrödinger’s equation with
an additional term:
∂ψ
(3.7)
Hψ + Dψ = i~ .
dt
The D operator is obtained by introducing equation (3.6) into here. The
calculations result
Dψ = i
e2 X
|~rik |2 (ωk − ωi )3 |hφk |ψi|2 |φi ihφi |ψi
24πc ik
(3.8)
The new term is neither linear nor invariant under transformation (2.2); it
is essentially irreversible.
The above sketched ideas may be far reaching and have to be investigated
from a great number of aspects but I hope they are not only the silly dreams
of an old man.
References
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WEB publication:
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