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Transcript
Path integral for the quantum harmonic oscillator using elementary methods
S. M. Cohen
Citation: American Journal of Physics 66, 537 (1998); doi: 10.1119/1.18896
View online: http://dx.doi.org/10.1119/1.18896
View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/66/6?ver=pdfcov
Published by the American Association of Physics Teachers
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Path integral for the quantum harmonic oscillator using
elementary methods
S. M. Cohen
Department of Physics, Portland State University, Portland, Oregon 97207
~Received 12 September 1997; accepted 12 November 1997!
We present a purely analytical method to calculate the propagator for the quantum harmonic
oscillator using Feynman’s path integral. Though the details of the calculation are involved, the
general approach uses only matrix diagonalization and well-known integrals, techniques which an
advanced undergraduate should understand. The full propagator, including both the prefactor and
the classical action, is obtained from a single calculation which involves the exact diagonalization
of the discretized action for the system. © 1998 American Association of Physics Teachers.
I. INTRODUCTION
Since their introduction,1 Feynman path integrals have become a powerful method of calculation for quantum mechanical problems.2,3 Though until recently exact solutions
were available for only the simplest cases, great advances in
developing methods of solving these integrals have been
made in the last 15 years.4 Yet even before these advances,
the approach bore fruit in many ways. For example, the
derivation5 of the ‘‘Feynman rules’’ was an extremely important contribution which greatly simplified calculations in
perturbation theory.
In a recent article, English and Winters6 have presented a
method of calculating the Feynman path integral for the prefactor of the propagator of the quantum harmonic oscillator.
The motivation for their work was ‘‘to introduce a formulation of quantum mechanics which is usually considered beyond the scope of most undergraduate courses.’’ We agree
with these authors that it is of interest to make alternative
approaches to quantum mechanics accessible to the undergraduate. We believe that path integrals have great beauty in
the simplicity of their basic formulation. They also clarify
various aspects of quantum mechanics, such as the uncertainty principle. The clarification in this particular case follows immediately from the central idea upon which the path
integral formulation is based: that all paths in configuration
space contribute to the evolution of the wave function. Thus
there is an intrinsic uncertainty as to the evolution of any
system ~we cannot know the trajectory the system follows!,
and this uncertainty is explicitly illustrated in this approach.
In this note, we give an alternative presentation which we
believe is somewhat more direct than that of English and
Winters. The method used by these authors required the use
of a symbolic computational program, and an intermediate
result written in terms of continued fractions. ~But see our
Appendix for a discussion of how the approach of these authors may be completed analytically.! Our method does not
require the use of a computer and is straightforward, formally, so it should be accessible to students. An understanding of Gaussian integrals, and of matrices and their eigenvectors and eigenvalues, are the only prerequisites to
following this approach.
Although this problem has been addressed in numerous
other works,2,3,6,7 our presentation is new in some important
ways. First, we discretize the action from the very beginning,
allowing us to obtain a final result which is exact for arbitrary N ~the number of intervals chosen for the
discretization—see below!. These results are thus directly
transferable to the case of a polymer chain with nonvanish537
Am. J. Phys. 66 ~6!, June 1998
ing bond lengths confined in a harmonic potential. Additionally, we show how the classical action arises naturally, along
with the prefactor, from a single calculation. This differs
from previous approaches in which only the prefactor was
calculated, the appearance of the classical action being assumed due to a theorem given by Feynman.2
II. FORMAL EVALUATION OF THE PATH
INTEGRAL
The quantum propagator, K(b,a) for a particle beginning
at position x(t a )5a and ending at x(t b )5b, is given as2
K ~ b,a ! 5
E
D @ x ~ t !# exp
S E
i
\
tb
ta
D
L @ x ~ t ! ,x˙ ~ t !# ,
~1!
where
L @ x ~ t ! ,x˙ ~ t !# 5 21 mx˙ 2 2 21 m v 2 x 2
~2!
is the classical Lagrangian, and the symbol * D @ x(t) # represents integration over all paths in configuration space beginning at a and ending at b. As is common practice, these
integrals may be done by first partitioning the time interval
into N pieces of width e each, so that T5t b 2t a 5N e . At the
end of the calculation the limits N→`, e →0, are taken,
such that T5N e is held constant. Then, with x j 5x( j e ), we
may write
K ~ b,a ! 5 lim
N→`
e →0
3
S
m
2 p i\ e
E E
`
2`
•••
D
N/2
`
2`
dx 1 dx 2 •••dx N21
N
3e im/2\ e ( j51 $ ~ x j 2x j21 !
22e2v2x2
j%
.
~3!
The argument of the exponential contains the quadratic form
N
Q5
(
j51
@~ x j 2x j21 ! 2 2 e 2 v 2 x 2j #
5x 20 1x 2N 2 e 2 v 2 x 2N 22x 1 x 0 22x N x N21 1Q 8 ,
~4!
where we may write, Q 8 5xW T AxW . Here,
xW T 5 ~ x 1 x 2 •••x N21 !
~5!
is the transpose of xW , and
© 1998 American Association of Physics Teachers
537
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A5
S
22 e 2 v 2
21
21
22 e v
0
0
0
•••
21
0
•••
21
22 e 2 v 2
21
•••
0
0
21
22 e 2 v 2
A
A
A
2
2
D
K ~ b,a ! 5 lim
N→`
e →0
.
~6!
(
j51
l j z 2j ,
~7!
K ~ b,a ! 5 lim
where the constant does not depend on the z j , then the
~coupled! integrals in Eq. ~3! will have been reduced to N
21 separate Gaussian integrals. First, we will find a transformation of variables,
~8!
N/2
exp
H F
GJ
im
x 2 1x 2 2 v 2 e 2 x 2N
2\ e 0 N
(
j51
FE S
`
)
j51
exp
2`
D G
im
l z 2 dz j .
2\ e j j
~14!
The integrals are now simple Gaussians, as advertised above,
yielding
N→`
e →0
xW 5OyW ,
D
~ ON21,j x N 1O1,j x 0 ! 2
lj
N21
3
N21
Q 8 5constant1
m
2 p i\ e
N21
2
If, using a change of variables from the x j to new variables
z j , we can rewrite Q 8 into the form
S
S
m
2 p i\ e
N21
2
(
j51
D H F
1/2
exp
im
x 2 1x 2 2 v 2 e 2 x 2N
2\ e 0 N
~ ON21,j x N 1O1,j x 0 ! 2
lj
5e i/\ S cl lim
N→`
e →0
S
m
2 p i\ e
D
GJ S ) D
N21
j51
21/2
lj
1/2
~ det A! 21/2[F ~ T ! e i/\ S cl.
~15!
such that
We will show below that
~9!
OT AO5L,
with L a diagonal matrix, L i j 5l j d i j , and O will be orthogonal since A is symmetric and real. Then we may write
S cl5 lim
N→`
e →0
N21
2
N21
Q5x 20 1x 2N 2 v 2 e 2 x 2N 22x N
N21
22x 0
(
j51
(
j51
ON21,j y j
(
j51
l j y 2j .
x N ON21,j 1x 0 O1,j
,
z j 5y j 2
lj
~11!
yielding
N21
(
j51
2
(
j51
`
2`
•••
`
2`
dx 1 dx 2 •••dx N21 ⇒
~12!
E E
`
2`
•••
`
2`
dz 1 dz 2 •••dz N21
~13!
in Eq. ~3! along with the transformation of variables. We
obtain
538
~16!
To implement the transformation of variables, Eq. ~8!, we
must find the matrix O which diagonalizes A. As is well
known, O is the matrix of the eigenvectors of A. It is not
difficult to show that a complete set of eigenvectors, which
we shall denote as eW j , is given in terms of their components
by
A S D
2
pi j
sin
,
N
N
~17!
with 1<i, j<N21. The corresponding eigenvalues are
~ ON21,j x N 1O1,j x 0 ! 2
.
lj
Since O is orthogonal, det O51, and the Jacobian of both
transformations, Eqs. ~8! and ~11!, is unity. Hence we have
the replacement
E E
G
III. DIAGONALIZATION OF THE MATRIX, A
~ eW j ! i 5
l j z 2j 1x 20 1x 2N 2 v 2 e 2 x 2N
N21
~ ON21,j x N 1O1,j x 0 ! 2
lj
~10!
Completing the squares, we change variables once again to
Q5
(
j51
is indeed the classical action, as it must be; and we will find
det A and thus the prefactor F(T), as well.
N21
O1,j y j 1
F
m
x 2 1x 2 2 v 2 e 2 x 2N
2e 0 N
Am. J. Phys., Vol. 66, No. 6, June 1998
l j 522 v 2 e 2 22 cos
S D
S D
pj
pj
54 sin2
2 v 2e 2,
N
2N
leading directly to the result
N21
det A5
N21
) l j 5 j51
)
j51
FS S D
)F
4 sin2
~ 24 ! N21
5 2
sin u 21
pj
2 v 2e 2
2N
DG
M /2
j51
sin2 u 2sin2
S DG
pj
M
,
~18!
~19!
where we have written sin u for v e /25 v T/2N, and M
52N. The product appearing in the final form of this equation is given in Hansen,8
S. M. Cohen
538
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M /2
)
j51
F
sin2 u 2sin2
S DG
pj
M
5 ~ 21 ! M /22 12M sin~ M u ! cot u ,
The evaluation of a 0 is identical to that of a N , apart from the
term proportional to 1/N; that is,
~20!
a 05
9
and we find
det A5
sin~ 2N u !
,
sin~ 2 u !
~21!
S cl~ N ! 5
F
mN 2 2 v 2 T 2 2
x 0 1x N 2 2 x N
2T
N
N21
2
5
~ ON21,j x N 1O1,j x 0 ! 2
lj
(
j51
G
mN
a N5
2T
S
O2N21,j
v 2T 2
12 2 2
.
N
lj
j51
(
But,
N21
(
j51
O2N21,j
lj
~23!
D
N21
5
5
5
2
N
2
N
N21
(
j51
(
j51
1
2N
4 sin2
M /221
(
j51
(
~24!
j51
S
D
S D
S D
S D
S D
S D
sin2
N21
(
j51
S D
S D
52
1
M
Mf
csc f csc
cos
2
2
SD
pj
N
pj
2sin2 u
2N
K N ~ b,a ! 5
~25!
S
HS D J
SD
S cl~ N ! 5
mN sin~ 2 u !
2 p i\T sin~ 2N u !
mN
2T
S
M
2k f
2
u 5arcsin
f
f
1
1
csc2
2 ~ 21 ! k sec2
,
4
2
4
2
~26!
D
v 2 T 2 cos~~ N22 ! f ! 2cos~ N f !
mN
a N5
12 2 2
.
2T
N
2 sin f sin~ N f !
Am. J. Phys., Vol. 66, No. 6, June 1998
~30!
D
1/2
e i/\ S cl~ N ! ,
HS
12
D
cos~~ N22 ! f ! 2cos~ N f ! 2
a
2 sin f sin~ N f !
D
v 2 T 2 cos~~ N22 ! f ! 2cos~ N f ! 2
2
b
N2
2 sin f sin~ N f !
J
2 sin f
ab ,
sin~ N f !
~32!
S D
vT
,
2N
~33!
and
S
D
v 2T 2
f 5arccos 12
.
2N 2
~27!
~31!
with
2
S
S D
S D
pj
N
52N sin f csc N f .
pj
cos f 2cos
N
~ 21 ! j sin2
1 12
where in our case we need k50,2. Then we have
539
(
j51
S D
S D
pj
N
mN sin f
52
.
pj
T sin~ N f !
cos f 2cos
N
~ 21 ! j sin2
Using the above results, we can determine the propagator
for an arbitrary number, N, of divisions of the time interval
T. This expression may be useful for students and others
doing numerical work with path integrals, as a check of their
discrete-time algorithms. The result is
4p j
12cos
M
,
2p j
cos f 2cos
M
2 p jk
M
2p j
cos f 2cos
M
cos
N21
ON21,j O1,j
lj
IV. EXACT PROPAGATOR FOR N DISCRETE
TIME INTERVALS
with, as before, M 52N, sin u5vT/2N, and also cos f51
2 v 2 T 2 /2N 2 . This sum may also be found in Hansen,10
M /221
m
T
(
j51
In writing the last equality, we have again referred to
Hansen,11
N21
p j ~ N21 !
N
p
j
4 sin2
2sin2 u
2N
sin2
~28!
~29!
5a N x 2N 1a 0 x 20 1a 0N x 0 x N .
Now,
N21
mN
a 0N 52
T
~22!
we may calculate
D
In the same way, we find
correct for all N> v T/2. Furthermore, taking
Oi j 5 ~ eW j ! i ,
S
mN
cos~~ N22 ! f ! 2cos~ N f !
12
.
2T
2 sin f sin~ N f !
~34!
Note that we have used x 0 5a, x N 5b.
Finally, the true propagator is obtained as
S. M. Cohen
539
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132.239.69.216 On: Wed, 15 Apr 2015 16:08:03
K ~ b,a ! 5 lim K N ~ b,a !
N→`
S
mv
5
2 p i\ sin v T
3exp
H
D
N21
)
j51
1/2
Aj
5
B j j51
J
)
im v
@~ a 2 1b 2 ! cos v T22ab # ,
2\ sin v T
5
~35!
Thus we have
having used the fact that for large N,
H
5
and
vT
.
N
In conclusion, we have presented a strictly analytical method
by which the full propagator for the quantum harmonic oscillator may be obtained using Feynman’s path integral approach. Though the details are involved, the general approach should be accessible to advanced students. In
particular, our presentation may be of interest to those instructors of graduate-level quantum mechanics who would
like to introduce path integrals into their courses.
5
5
S
m
2 p i\T
D
S
m
2 p i\T
D
S
m
2 p i\T
DH
F~ T !5
S
J
1/2
lim
N→`
1/2
lim
N→`
1/2
H
H
~A5!
F S
DG
DG
v 2T 2
2N 2
N
v 2T 2
sin N arccos 12
2N 2
sin arccos 12
F
F G
F S DG
vT
N
N
vT
sin N
N
vT
sin~ v T !
sin
J
Finally, then
ACKNOWLEDGMENTS
We would like to thank Pui-Tak Leung for helpful comments and for bringing Ref. 6 to our attention.
[email protected] j arccos~ g /2 !#
[email protected] arccos~ g /2 !#
[email protected]~ j11 ! arccos~ g /2 !#
[email protected] arccos~ g /2 !#
[email protected] arccos~ g /2 !#
.
[email protected] N arccos~ g /2 !#
F~ T !
vT
,
u>
2N
f>
N21
mv
2 p i\ sin~ v T !
S
J
J
1/2
1/2
1/2
~A6!
.
D
1/2
~A7!
.
This is the desired result, once again obtained by purely analytical means.
APPENDIX
1
In Ref. 6, it is shown that the prefactor, F(T), may be
written as a product of factors,
S
m
F~ T !5
2 p i\T
D
1/2
F
lim N
N→`
N21
)
j51
Aj
Bj
G
1/2
,
~A1!
where the A j and B j satisfy the same recursion relations,
A j 5 g A j21 2A j22 ,
~A2!
B j 5 g B j21 2B j22 ,
~A3!
with g 522 v 2 T 2 /N 2 , and starting conditions, A 21 521,
A 0 50, B 21 50, and B 0 51. As these authors observe, A j11
5B j . What we would like to point out is that these relations
brand these objects as Chebyshev polynomials of the second
kind.12 Specifically,
B j 5U j
SD
g
[email protected]~ j11 ! arccos~ g /2 !#
5
.
2
[email protected] arccos~ g /2 !#
~A4!
Therefore, the product appearing in the formula for F(T)
is just
540
Am. J. Phys., Vol. 66, No. 6, June 1998
R. P. Feynman, ‘‘Space-time approach to nonrelativistic quantum mechanics,’’ Rev. Mod. Phys. 20, 367–387 ~1948!.
2
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals
~McGraw–Hill, New York, 1965!.
3
L. S. Schulman, Techniques and Applications of Path Integration ~WileyInternational, New York, 1981!.
4
C. Grosche and F. Steiner, ‘‘How to solve path integrals in quantum mechanics,’’ J. Math. Phys. 36, 2354–2385 ~1995!, and references therein.
5
R. P. Feynman, ‘‘An operator calculus having applications in quantum
electrodynamics,’’ Phys. Rev. 84, 108–128 ~1951!.
6
L. Q. English and R. R. Winters, ‘‘Continued fractions and the harmonic
oscillator using Feynman’s path integrals,’’ Am. J. Phys. 65, 390–393
~1997!.
7
J. T. Marshall and J. L. Pell, ‘‘Path-integral evaluation of the space-time
propagator for quadratic Hamiltonian systems,’’ J. Math. Phys. 20, 1297–
1302 ~1979!.
8
Eldon R. Hansen, A Table of Series and Products ~Prentice-Hall, Englewood Cliffs, NJ, 1975!, p. 497, Eq. #91.1.18.
9
This result has been obtained previously in H. Kleinert, Path Integrals in
Quantum Mechanics, Statistics, and Polymer Physics ~World Scientific,
Singapore, 1990!, p. 85, Eq. #2.140.
10
Reference 8, p. 272, Eq. #41.2.19.
11
Reference 8, p. 272, Eq. #41.2.25.
12
I. S. Gradshteyn and I. M. Rhyzik, Table of Integrals, Series, and Products
~Academic, Orlando, FL, 1980!, pp. 1032–1033.
S. M. Cohen
540
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