Download Quantum mechanical approaches to the virial S.LeBohec

Quantum mechanical approaches to the virial
S.LeBohec
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA
(Dated: June 30th 2015)
In this note, we approach the virial from a standard quantum mechanics point of view. Section
1 reviews the classical virial theorem. Section 2 reviews the Ehrenfest theorem as we will use it
in the quantum discussion. Then the virial is considered quantum mechanically in two different
ways. In section 3, the expectation values of the position and momentum observables considered
independently are used to construct a ”classical” virial for which we derive the virial theorem and
thereby establish a property of wave function. In section 4, we define a quantum virial observable
and establish the quantum virial theorem. Section 5 is a brief discussion and conclusion highlighting
the fact the quantum virial theorem is in direct correspondence with the classical virial theorem
while the ”classical” virial considered from a quantum mechanical approach merely corresponds to
an integral property of the wave functions.
I.
VIRIAL THEOREM IN CLASSICAL MECHANICS
The virial is a quantity that arises from considering the time derivative of the moment of inertia I about the origin
for a system of particles. Take a system of N particles, each with a mass mi , position ri and momentum pi with i
N
P
running from 1 to N . Then I =
mi |ri |2 and
i=1
N
N
X
X
dri
dI
2mi ri ·
ri · pi = 2G
=
=2
dt
dt
i=1
k=1
where G =
N
P
ri · pi defines the virial.
i=1
We can then look at the time derivative of the virial:
N
N
N
X
X
dpi
dG X dri
ri ·
r i · Fi
=
· pi +
= 2T +
dt
dt
dt
i=1
i=1
i=1
where we introduced T the kinetic energy of the entire system and the force Fi acting on the ith particle .
The virial theorem derives from considering the time average of the time derivative of the virial. We denote the
average over a time period τ as (· · · )τ .
N
P
In a reference frame where the the system is globally at rest,
pi = 0, a finite and stable bound system would be
i=1
such that ri and pi are bound, so, in the limit of infinite times, the time average dG
dt ∞ = 0. Consequently,
2(T )∞ = −
N
X
i=1
!
ri · Fi
(1)
∞
which express the virial theorem [5] in its general form.
We may then consider each particle to only be under the influence of a superposition of pairwise interactions with
N
P
every other particle. The force exerted by all the particles of the system on the ith particle is Fi =
Fij where
j=1;j6=i
Fij is the force exerted by particle j on particle i. With this, we can write
N
X
i=1
ri · Fi =
N X
X
i=1 j<i
where we used [6] the fact that Fij = −Fji .
ri · Fij +
N X
X
i=1 j>i
ri · Fij =
N X
X
(ri − rj ) · Fij
i=1 j<i
(2)
2
We now further restrict ourselves to the cases in which the force between any two particles derives from a central
potential V (|r|):
Fij = −∇i V (|ri − rj |) = −
Using this in the above result, we obtain:
!
N P
P
dV
rij dr (rij ) .
2(T )∞ =
N
P
i=1
r i · Fi = −
dV ri − rj
dV ri − rj
=−
dr |ri − rj |
dr rij
N P
P
i=1 j<i
rij dV
dr . So we have, for the virial theorem, the form:
i=1 j<i
Furthermore, if the interaction potential energy is proportional to a power law of the distance V (r) = V0 · rν , then,
N P
P
V (rij ), the virial theorem takes its most usual
introducing the total potential energy of the system VT ot =
i=1 j<i
form:
2(T )∞ = ν(VT ot )∞
(3)
This can be applied with ν = −1 in the case in the case of a Keplerian potential or with ν = 2 in the case of a network
of harmonic oscillators.
II.
THE EHRENFEST THEOREM
The virial theorem discussed in the previous section concerns time averaging (· · · )∞ in the limit of infinite times.
In the quantum discussion of the properties of the virial, this need to be combined with the statistical nature of
the outcome of measurements in quantum mechanics. As a consequence, we are going to concentrate on the time
evolution of expectation values. Considering an observable A, the expectation value of this observable is denoted
hAi = hφ(t)|A|φ(t)i when the considered system is in the quantum state |φ(t)i. The Ehrenfest theorem provides an
expression for the time derivative of expectation values. In order to establish this expression, we can proceed directly
from the definition of hAi:
d
d
d
∂A
hAi =
hφ(t)| A|φ(t)i + hφ(t)|A
|φ(t)i + hφ(t)|
|φ(t)i
dt
dt
dt
∂t
We assumed A includes an explicit time dependence. If the time evolution of the state of the system is governed by
d
d
the Schrödinger equation with a Hamiltonian H, then, i~ dt
|φi = H|φi and, since H is hermitian, i~ dt
hφ| = −hφ|H.
So we obtain the expression of the Ehrenfest theorem:
d
1
∂A
hAi = hφ(t)|AH − HA|φ(t)i + hφ(t)|
|φ(t)i
dt
i~
∂t
or, using the usual notations for the commutator and the expectation value,
d
1
∂A
hAi = h[A, H]i + h
i
dt
i~
∂t
Particularly interesting applications of the Ehrenfest theorem appear when considering position A = R and momentum A = P operators. Consider a particle of mass m whose evolution is governed by a Hamiltonian H = P 2 /2m + V
where V is the potential energy. In order to apply the Ehrenfest theorem, we need to express the commutator [R, H]
and [P, H]. This can be done using [R, P ] = i~ :
1
1
[R, P 2 ] =
RP 2 − P 2 R
2m
2m
P
1
((i~ + P R) P − P (RP − i~)) = i~
=
2m
m
[R, H] =
We also need to express [P, H] = [P, V ], which can be done in position representation with P =
[P, H] = [P, V ] =
~
~
~
(∇V − V ∇) = ((∇V ) + V ∇ − V ∇) = (∇V )
i
i
i
~ ∂
i ∂R
= ~i ∇:
3
Since R and P do not have any explicit time dependence, the Ehrenfest theorem then directly gives the two following
relations for the ith particle of the quantum analog of the classical system considered in our discussion of the virial
theorem in section I:
d
Pi
hRi i = h i
dt
m
d
hPi i = −h∇i VT ot i = hFi i
dt
(4)
(5)
These are Hamilton’s equations in which we re-introduced Fi , the force acting on particle i. The time derivative of
the expected values of the positions are equal to the expectation values of momenta divided by the mass. The time
derivatives of the expectation values momenta are equal to the expectation values of the forces. This is an important
result as it provides a bridge between the quantum and classical regimes. It establishes that the time evolution of
expectation values in Born’s probabilistic interpretation of quantum mechanics matches the prescriptions of classical
mechanics.
III.
CLASSICAL VIRIAL THEOREM IN THE QUANTUM REGIME
The state of a system of N distinguishable particles can be described by the direct product of the wave functions
N
P
of its individual constituents. We can then classically define the virial as GC =
hPi ihRi i with Ri and Pi the
i=1
position and momentum operators for particle i. This virial can be regarded as classical since, following the Ehrenfest
theorem, hRi i and hPi i in the quantum system will evolve with time in exactly the same way as ri and pi in the
classical system for which we have established the virial theorem in Section I. In particular, we already know that
the classical virial theorem (Equation 1) directly applies:
N
X
hPi i2
i=1
!
=
mi
!
N
X
hRi i · hFi i
i=1
∞

=
∞
N X
X

(hRi i − hRj i) · hFij i
i=1 j<i
∞
If the force Fij derives from a central potential of the form V0 |Ri − Rj |ν , this gives.
N
X
hPi i2
i=1
mi
!


N X
X
= νV0 
(hRi i − hRj i) · h|Ri − Rj |ν−2 (Ri − Rj )i
i=1 j<i
∞
(6)
∞
But still, let us follow the derivation as an exercise. Noting the components k ∈ {x, y, z} of the position and
momentum of the ith particle as Rik and Pik respectively, we can take the time derivative of GC :
N
XX
d
GC =
dt
i=1
k
d k
d k
hPi i hRik i + hPik i
hRi i
dt
dt
We can then apply the Ehrenfest theorem:
N
d
1 XX
GC =
h[Pik , H]ihRik i + hPik ih[Rik , H]i
dt
i~ i=1
k
And, using the expressions we found for the commutators [R, H] and [P, H] in Section II
N X
N
X
X
∂VT ot
hPi i2
dGC
=−
hRik ih
i
+
dt
mi
∂Rik
i=1
i=1
k
Where we make use of the total potential energy operator,
VT ot =
N X
X
l=1 j<l
N
V (|Rl − Rj |) =
1 XX
V (|Rl − Rj |)
2
l=1 j6=l
4
and in the case of the power law central potential:
X
∂VT ot
= νV0
|Ri − Rj |ν−2 (Rik − Rjk )
k
∂Ri
j6=i
Time averaging for a bound system in the reference frame where it is at rest
N
X
hPi i2
i=1
mi
!
dGC
dt ∞
= 0, we obtain


N XX
X
= νV0 
hRik ih|Ri − Rj |ν−2 (Rik − Rjk )i
i=1 j6=i
∞
k
(7)
∞
Using the same manipulation as in Section I (see Equation 2), we find this is equivalent to Equation 8 and we have
completed a quantum mechanical derivation of the classical virial theorem.


!
N
N X
X
X
hPi i2
= νV0 
(hRi i − hRj i) · h|Ri − Rj |ν−2 (Ri − Rj )i
(8)
m
i
i=1
i=1 j<i
∞
∞
It should be noted that the lefthand side is not the time averaged expectation value of the kinetic energy. It is the
time averaged kinetic energy for the expectation value of the momenta. In the same way, the right hand side can
not be written simply in terms of the total potential energy so the classical virial theorem can not be expressed
quantum-mechanically in a form similar to Equation 3.
In this equation, it should be highlighted that the expectation values h· · · i are to be understood as calculated for
the many particle quantum state |φ(t)i of the system in the course of its evolution following Schrödinger’s equation.
This quantum form of the classical virial theorem therefore stands as a non-trivial property of the solution of the
Schrödinger equation.
IV.
QUANTUM VIRIAL THEOREM
We could also consider the expectation value of the quantum virial GQ =
N P
P
i=1 k
hPik ·Rik i. This is the usual approach
to the virial theorem in quantum mechanics as originally investigated by Vladimir Fock [3] in 1930. Alternatively,
N P
P
we could consider G†Q =
hRik · Pik i. However, as long as we are interested only in the time derivative of the
i=1 k
expectation value of GQ (or G†Q ), this makes no difference. Indeed:
N
N
dG†Q
dGQ
d XX k k
d XX
=
hPi Ri i =
hRik · Pik i − i~ =
dt
dt i=1
dt i=1
dt
k
k
Since there are no explicit time dependences, the Ehrenfest theorem gives:
N
1 XX k
dGQ
=
h[Pi · Rik , H]i
dt
i~ i=1
k
Considering the same Hamiltonian as in the previous section, we see that we need to calculate [GQ , P 2 ]
2
2
3
2
2
[Pik Rik , Pik ] = Pik Rik Pik − Pik Rik = Pik (Rik Pik − Pik Rik ) = 2i~Pik
2
and [GQ , V ] can be obtained in position representation:
∂
∂
~
k
k
R
V
−
V
R
[Pik Rik , VT ot ] =
T ot
T ot
i ∂Rik i
∂Rik i
~
∂V
∂
k
k ∂
=
VT ot + Rik
+
R
V
−
V
−
V
R
T ot
T ot i
i T ot
i
∂Rik
∂Rik
∂Rik
~
∂V
= Rik ·
i
∂Rik
5
so
2
[Pik Rik , H] = i~
∂VT ot
Pik
− i~Rik
mi
∂Rik
with this:
2
N
N
X X hP k i X X
dGQ
∂VT ot
i
i
=
−
hRik
dt
m
∂Rik
i
i=1
i=1
k
k
Here we recognize the first term as twice the expectation value of the system kinetic energy T =
N P
2
P
hPik i
. At the
dG
same time, considering the time average over an infinite period, a bound system in its rest frame satisfies dtQ
=0
∞
N
P P k ∂VT ot
hRi ∂Rk i
.
so that 2(T )∞ =
i=1 k
2mi
i=1 k
i
∞
In cases where the potential energy is a power law of index ν:


N XX
k − Rk
X
R
V
(|R
−
R
|)
i
j
j
i
2(T )∞ = ν
hRik
i
|R
−
R
|
|R
−
R
|
i
j
i
j
i=1
j6=i
k
∞
or

N X
2
X
V (|Ri − Rj |) |Ri | − Ri · Rj 
h
i
= ν
|Ri − Rj |
|Ri − Rj |
i=1

2(T )∞
j6=i
∞
or

2(T )∞

N X
2 − 2R · R + |R |2
X
V
(|R
−
R
|)
|R
|
i
j
i
i
j
j
= ν
h
i
|R
−
R
|
|R
−
R
|
i
j
i
j
i=1 j<i
∞
or

N X
X
hV (|Ri − Rj |)i
= ν

2(T )∞
i=1 j<i
∞
or, finally,
2hT i∞ = νhVT ot i∞
and we recover the virial theorem in the exact same formulation as in classical mechanics except for the fact that the
kinetic and potential energies have to be replaced by their expectation values.
V.
SUMMARY AND CONCLUSIONS
We have defined the virial for a quantum system in two different ways.
N
P
In Section III, we considered a classical virial, GC =
hPi ihRi i. Since hPi i and hRi i have the same equation of
i=1
motion as the corresponding quantities in the classical system, we could apply the classical virial theorem directly to
the quantum form replacing ri , pi and Fi by their expectation values. We were then able to derive the same relation
quantum-mechanically:


!
N
N X
X
X
hPi i2
= νV0 
(hRi i − hRj i) · h|Ri − Rj |ν−2 (Ri − Rj )i
m
i
i=1
i=1 j<i
∞
∞
6
The classical virial theorem can be seen as an integral property of the solutions of the Schrödingier equation.
N
P
hPi · Ri i. We obtained a virial theorem
In Section IV, we considered the more usual quantum virial, GQ =
i=1
with exactly the same form as in classical mechanics provided the kinetic and potential energies are replaced by their
quantum mechanical expectation values:
2(hT i)∞ = ν(hVT ot i)∞
This suggests that the operation of taking the expectation value h· · · i can be regarded as a continuation of the time
averaging (· · · )τ to reveal the contribution of a dynamics internal to the wave function. In fact, when considering
the system to be in a stationary state, the time averaging becomes superfluous and we obtain a relation between
expectation values 2hT i = νhVT ot i which, when regarded as the time averaging of an internal dynamics, is identical
to the classical form of the virial theorem: 2(T )∞ = ν(VT ot )∞ . At the same time and in complementarity, the virial
theorem for GC becomes degenerate since there are no dynamics other than those internal to the wave function,
leading to hPi i = 0.
This consideration of the virial theorem in quantum mechanics certainly does not provide any proof for the necessity
of an explicit continuation between the classical dynamic dominating at scales larger than the de Broglie wavelength
and a particulate dynamic internal to the wave function. However, the fact that the classical (Section I) and quantum
(Section IV) virial theorems are identical in form, follows well from Nelson’s stochastic quantization [2] and more
recently from Nottale’s scale relativity [4]. From these point of views, the wave function property revealed by the
quantum consideration of the classical virial in Section III can be seen as a superfluous matching constraint appearing
because of the artificial discrimination between the dynamics at classical and quantum scales, with the wave function
merely playing the role of a wrapper of the latter.
VI.
ACKNOWLEDGEMENT
In the preparation of this note, I have used the Wikipedia page on the virial theorem
(https://en.wikipedia.org/wiki/Virial theorem). I am grateful to Janvida Rou for her helpful comments and
to Eugene Mishchenko for noticing an error of reasoning in an earlier version of this note.
[1]
[2]
[3]
[4]
L.F.Abbott & M.B.Wise, ”Dimension of a quantum-mechanical path”, Am.J.Phys. 49(1), 37-39, 1981
E.Nelson, ”Derivation of the Schrdinger Equation from Newtonian Mechanics”, Phys. Rev. 150, 1079 (1966)
V.Fock, ”Bemerkung zum Virialsatz”. Zeitschrift fr Physik A 63 (11): 855858 (1930)
L.Nottale, ”Scale Relativity And Fractal Space-Time: A New Approach to Unifying Relativity and Quantum Mechanics”;
World Scientific Publishing Company; 1 edition, 2011; ISBN 978-1848166509
[5] The word ”virial” derives from latin ”vis” which mean force. The word and the theorem are both due to Rudolf Clausius
in 1870 in ”On a Mechanical Theorem Applicable to Heat”, in Philosophical Magazine, Ser. 4, vol. 40, 1870, p. 122127
[6] Using the fact that Fij = −Fji , we have:
N
X
i=1
ri · Fi =
N X
X
ri · Fij +
N X
X
i=1 j<i
ri · Fij =
i=1 j>i
N X
X
ri · Fij −
i=1 j<i
N X
X
ri · Fji
i=1 j>i
We can write the last term regrouping those obtained for the same values of j:
N X
X
ri · Fji = (r1 · F21 ) + (r1 · F31 + r2 · F32 ) + (r1 · F41 + r2 · F42 + r3 · F43 ) + · · ·
i=1 j>i
and we see that
N X
X
ri · Fji =
i=1 j>i
N X
X
ri · Fji =
j=1 i<j
N X
X
i=1 j<i
Combining the two terms in the original expression:
N
X
i=1
ri · Fi =
N X
X
(ri − rj ) · Fij
i=1 j<i
rj · Fij