Download The variational principle and simple properties of the ground

The variational principle and simple properties of the ground-state
wave function
J. Mur-Petita) and A. Polls
Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona, Avda. Diagonal 647,
E-08028 Barcelona, Spain
F. Mazzanti
Departament d’Electrònica, Enginyeria i Arquitectura La Salle, Universitat Ramon Llull, Pg. Bonanova 8,
E-08022 Barcelona, Spain
共Received 8 October 2001; accepted 28 March 2002兲
The variational principle is used to show that the ground-state wave function of a one-body
Schrödinger equation with a real potential is real, does not change sign, and is nondegenerate. As a
consequence, if the Hamiltonian is invariant under rotations and parity transformations, the ground
state must have positive parity and zero angular momentum. © 2002 American Association of Physics
Teachers.
关DOI: 10.1119/1.1479742兴
I. INTRODUCTION
II. REALITY AND POSITIVE DEFINITENESS
The variational method is considered to be one of the most
useful tools for finding approximate solutions of the Schrödinger equation. Usually it is presented in introductory
courses on quantum mechanics as an easy way to find good
estimates of the ground-state wave function. After stating
that the expectation value of the Hamiltonian for any suitable
trial wave function ⌿,
We begin by considering the stationary Schrödinger equation for a spinless particle in three dimensions under the
influence of a real potential V(r),
具H典⫽
具⌿兩H兩⌿典
,
具⌿兩⌿典
Am. J. Phys. 70 共8兲, August 2002
ប2 2
ⵜ ⌿ 共 r兲 ⫹V 共 r兲 ⌿ 共 r兲 ⫽E⌿ 共 r兲 ,
2m
共2兲
共1兲
provides an upper bound to the ground-state energy, E gs , the
students are given several exercises where the minimization
for a given Hamiltonian, for some family of functions, allows them to find the exact solution. Obviously, the minimization of the expectation value of the Hamiltonian within the
full Hilbert space is equivalent to the exact resolution of the
stationary Schrödinger equation. In fact, the power of the
method goes far beyond this simple picture, as has been
shown for example in the study of strongly interacting quantum fluids, where the optimization of a trial wave function
allows for the proper description of the ground state of the
system.1
Here we propose a simple way to illustrate the power of
the variational principle by deriving some general properties
of the ground-state wave function of a one-body Hamiltonian
corresponding to a particle with no internal degrees of freedom and moving in a field defined by a potential V(r). In
particular, it is shown that the ground-state wave function
can be taken to be real and non-negative, and that it cannot
be degenerate. Other consequences for the angular momentum and the parity of the ground state are also presented.
There is a vast literature on the properties of the groundstate wave function for very general potentials. For the
simple case studied here, other more elaborated proofs can
be found in classical textbooks,2 where a complete analysis
based on the general properties of the solutions of Sturm–
Liouville problems is presented. Other methods, based on the
path-integral formulation of quantum mechanics and
Feynman–Kac’s formula have also been used to analyze this
particular type of potential.3
808
H⌿ 共 r兲 ⫽⫺
http://ojps.aip.org/ajp/
where H is the Hamiltonian and E a given eigenvalue. We
restrict ourselves to this case, because velocity-dependent
potentials can give rise to spontaneous symmetry breaking,
as pointed out in Ref. 4. Spontaneous symmetry breaking
could lead to degeneracy in the ground-state wave function
and the proof given here does not hold. If the Hamiltonian is
rotationally invariant, the potential depends only on r and
thus V(r)⫽V(r).
Let us assume that H has at least one bound state. The
most general trial wave function can be written in the following form:
⌿ T 共 r兲 ⫽ f 共 r兲 e i ␹ (r) ,
共3兲
where both f and ␹ are real-valued functions, and f ⭓0. The
next step is to evaluate the expectation value of the Hamiltonian. The potential energy is easily found and is given by
E pot⫽
具 ⌿ T 兩 V 兩 ⌿ T 典 兰 d 3 r f 2 共 r兲 V 共 r兲
⫽
,
兰 d 3 r f 2 共 r兲
具 ⌿ T兩 ⌿ T典
共4兲
and therefore is not affected by the possible complex character of the wave function.
If we take into account the anti-Hermitian nature of the
gradient operator, the kinetic energy can be calculated using
the Jackson-Feenberg identity5
© 2002 American Association of Physics Teachers
808
ប 2 兰 d 3 r 关共 ⵜ 2 ⌿ T* 兲 ⌿ T ⫹⌿ T* 共 ⵜ 2 ⌿ T 兲 ⫺2 共 ⵜ⌿ T* 兲 • 共 ⵜ⌿ T 兲兴
具 ⌿ T兩 T 兩 ⌿ T典
⫽⫺
,
8m
兰 d 3 r⌿ T* ⌿ T
具 ⌿ T兩 ⌿ T典
具 ⌿ T兩 T 兩 ⌿ T典
具 ⌿ T兩 ⌿ T典
easily derived. For instance, if the Hamiltonian is invariant
under rotations, it commutes with the angular momentum
operator and hence both operators can be diagonalized in the
same basis. Therefore, their common eigenstates may be
characterized by the eigenvalues corresponding to both operators. Because each state with angular momentum l is 2l
⫹1 times degenerate, we readily conclude that the angular
momentum of the ground state has to be zero. Furthermore,
if the Hamiltonian is invariant under parity transformations,
that is, under the change r→⫺r, the positive character of the
ground state implies that the corresponding wave function
has positive parity, ⌿ 0 (r)⫽⌿ 0 (⫺r).
ប 2 兰 d 3 r 共 ⵜ f 共 r兲兲 2 ប 2 兰 d 3 r f 2 共 r兲共 ⵜ ␹ 共 r兲兲 2
⫹
2m 兰 d 3 r f 2 共 r兲
2m
兰 d 3 r f 2 共 r兲
V. APPLICATION TO THE HARMONIC
OSCILLATOR
which has been used mostly in the context of quantum
liquids.6 It is easy to see that
ⵜ⌿ T ⫽e i ␹ 关 ⵜ f ⫹i f ⵜ ␹ 兴 ,
共6兲
and therefore
共 ⵜ 2 ⌿ T* 兲 ⌿ T ⫹⌿ T* 共 ⵜ 2 ⌿ T 兲 ⫽2 f ⵜ 2 f ⫺2 f 2 共 ⵜ ␹ 兲 2 ,
共7兲
ⵜ⌿ T* •ⵜ⌿ T ⫽ 共 ⵜ f 兲 2 ⫹ f 2 共 ⵜ ␹ 兲 2 ,
共8兲
which leads to
E kin⫽
⫽
f
␹
⬅E kin
⫹E kin
,
f
␹
⫹E kin
⫹E pot .
具 H 典 ⫽E kin
共9兲
共10兲
␹
E kin
is the only part of 具 H 典 that depends on ␹.
The quantity
␹
Because E kin
⭓0, we can always decrease the energy associated with the trial wave function 共3兲 by setting
ⵜ ␹ 共 r兲 ⫽0,
共11兲
that is, by making the phase a constant. As a global phase in
the wave function cannot affect the results, we can always
choose ␹ ⬅0. As a consequence, the ground-state wave function should always admit the following representation:
⌿ T 共 r兲 ⫽ f 共 r兲 ⭓0,
共12兲
thus indicating that it is real and does not change sign. This
proof fails for velocity-dependent potentials because in that
case Eq. 共4兲 does not hold.
III. NONDEGENERACY
We can prove the nondegeneracy of the ground state by
assuming that it is degenerate and arriving at a contradiction.
Suppose that there exists a wave function ⌿̃ T 共apart from
⌿ T 兲 with the same ground-state energy E T . This wave function should be orthogonal to ⌿ T ,
具 ⌿ T 兩 ⌿̃ T 典 ⫽
冕
d 3 r⌿ T* 共 r兲 ⌿̃ T 共 r兲 ⫽0,
共13兲
but due to the fact that ⌿ T is positive everywhere and condition 共13兲, ⌿̃ T should change sign at some point, contrary to
what has been shown previously. Therefore, we conclude
that the presumed hypothesis is incorrect, and that the
ground-state wave function cannot be degenerate.
IV. ANGULAR MOMENTUM AND PARITY
Up to this point, the arguments presented here apply
equally well for any spatial dimension. An equivalent proof
has been used to demonstrate similar properties of the
ground-state wave function of a system of interacting
bosons.7 Moreover, several interesting consequences can be
809
共5兲
Am. J. Phys., Vol. 70, No. 8, August 2002
In this section, the variational method is applied to the
harmonic oscillator in three dimensions. First, we follow the
traditional way found in quantum mechanics textbooks.8,9
We assume a trial ground-state wave function, calculate the
corresponding energy, minimize it, and then compare the resulting estimate with the exact result given by
⌿ gs共 r兲 ⫽
冉 冊
m␻
␲ប
3/4
e ⫺m ␻ r
2 /2ប
共14兲
,
E gs⫽ 23 ប ␻ ,
共15兲
where ␻ is the oscillator frequency appearing in the potential
V⫽ 12 m ␻ 2 r 2 .
Suppose our intuition leads us to think that a Gaussian
trial wave function would be a good choice
2
⌿ T 共 r兲 ⫽Ne ⫺ ␣ r /2.
共16兲
Direct application of the variational method will determine
the optimum value of ␣, that is, the value that minimizes the
expectation value of the Hamiltonian. To do so, we find the
value of N from the normalization condition
具 ⌿ T 兩 ⌿ T 典 ⫽1⇒N⫽
冉冊
␣
␲
3/4
共17兲
,
while the kinetic and potential energies are given by
ⵜ 2 ⌿ T 共 r 兲 ⫽⫺ ␣ 共 3⫺ ␣ r 2 兲 ⌿ T 共 r 兲 ⇒E kin⫽
3 ប2
␣,
4 m
E pot⫽ 具 21 m ␻ 2 r 2 典 ⫽ 43 m ␻ 2 ␣ ⫺1 .
共18兲
共19兲
In this way, the total energy for the trial wave function is the
sum of the last two terms, and becomes a function of the
parameter ␣,
具 H 典 ⬅E 共 ␣ 兲 ⫽
3 ប2
3
␣ ⫹ m ␻ 2 ␣ ⫺1 .
4 m
4
共20兲
The value of ␣ that minimizes E( ␣ ) is
dE
d␣
冏
⫽0⇒ ␣ T ⫽
␣⫽␣T
m␻
1
⫽ 2,
ប
␭
Mur-Petit, Polls, and Mazzanti
共21兲
809
where ␭⫽ 冑ប/m ␻ is the length scale of the harmonic oscillator potential. If we substitute this result for ␣ T into E( ␣ ),
the minimum energy corresponding to the family of functions associated with our ansatz 共16兲 becomes
E T ⫽E 共 ␣ T 兲 ⫽ 23 ប ␻ .
共22兲
We recover the exact result 共15兲 because the trial wave function has the correct functional dependence on r.
If the exact functional form is unknown, as is usually the
case, one can still use the information provided by the variational principle: the ground state should be real and nonnegative. This information can be incorporated into the trial
wave function by expressing it in the following way:
⌿ T 共 r兲 ⫽ f 共 r兲 ⫽e
⫺ ␾ (r)
,
共23兲
with ␾ (r)苸R. Because the ground-state wave function must
be nondegenerate, we can assume that ␾ (r)⫽ ␾ (r), so that it
has zero angular momentum. If we substitute this ansatz into
the Schrödinger equation and do a little algebra, we obtain a
differential equation for ␾ (r),
␾ ⬘ 共 r 兲 r 2 2mE
␾ ⬙ 共 r 兲 ⫺ 共 ␾ ⬘ 共 r 兲兲 2 ⫹2
⫹ 2⫽ 2 ,
r
␭
ប
共24兲
where the prime denotes derivative with respect to r. From
the variational point of view, Eq. 共24兲 can be obtained by
requiring ␾ to be optimized, that is, by imposing the expectation value of the Hamiltonian in the trial wave function
共23兲 to be stationary with respect to small variations in ␾
␦
关 具 ⌿ T 兩 H 兩 ⌿ T 典 ⫺E 具 ⌿ T 兩 ⌿ T 典 兴 ⫽0,
␦␾
共25兲
where E plays the role of a Lagrange multiplier introduced to
keep the normalization condition.
The simplest regular solution of Eq. 共24兲 is
␾共 r 兲⫽ ␣ Tr 2
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Am. J. Phys., Vol. 70, No. 8, August 2002
共26兲
with ␣ T ⫽1/2␭ 2 , thus leading to the exact solution 共14兲.
In summary, we have stressed the power of the variational
method by deriving general properties of the ground-state
wave function of a one-body Hamiltonian in a simple way.
This information can be used to constrain the form of a trial
wave function when looking for solutions of the Schrödinger
equation. For the three-dimensional harmonic oscillator, we
have shown that the information provided by the variational
method is enough to find the exact solution of the problem.
ACKNOWLEDGMENTS
The authors are grateful to Professor Rolf Tarrach and
Professor Josep Taron for useful discussions. This work has
been partially supported by DGICYT 共Spain兲 Grant No.
PB98-1247, and the Program No. 1998SGR00011 from the
Generalitat de Catalunya. One of us 共J.M.兲 wishes to acknowledge support from a fellowship of the Fundació Universitària Agustı́ Pedro i Pons.
a兲
Electronic mail: [email protected]
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