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H. Kleinert, PATH INTEGRALSJanuary 6, 2016 (/home/kleinert/kleinert/books/pathis/pthic9.tex)

Tollimur in caelum curvato gurgite, et idem

subducta ad manes imos descendimus unda

We are carried up to the heaven by the circling wave,

and immediately the wave subsiding, we descend to the lowest depthsVirgil (70BC–19BC), Aeneid, 3, 564

9

Wave Functions

The fundamental quantity obtained by solving a path integral is the time evolutionamplitude or propagator of a system (xbtb|xata). In Schrodinger quantum mechanics,on the other hand, one has direct access on the energy spectrum and the wavefunctions of a system [see (1.94)]. This chapter will explain how to extract thisinformation from the time evolution amplitude (xbtb|xata). The crucial quantityfor this purpose the Fourier transform of (xbtb|xata), the fixed-energy amplitudeintroduced in Eq. (1.317):

(xb|xa)E =∫ ∞

tadtb exp {iE(tb − ta)/h} (xbtb|xbta), (9.1)

which contains as much information on the system as (xbtb|xata), and gives, inparticular, a direct access to the energy spectrum and the wave functions of thesystem. This is done via the the spectral decomposition (1.325).

Alternatively, we can work with the causal propagator at imaginary time,

(xbτb|xaτa) =∫

dDp

(2πh)Dexp

[

i

hp(xb − xa)−

p2

2Mh(τb − τa)

]

, (9.2)

and calculate the fixed-energy amplitude by the Laplace transformation

(xb|xa)E = −i∫ ∞

τadτb exp {E(τb − τa)/h} (xbτb|xbτa). (9.3)

9.1 Free Particle in D Dimensions

For a free particle in D dimensions, the fixed-energy amplitude was calculated inEqs. (1.348) and (1.355). It will be instructive to rederive the same result once moreusing the development in Section 8.5.1. Here we start directly from the spectralrepresentation (1.325), which for a free particle takes the explicit form Eq. (1.344):

(xb|xa)E =∫

dDk

(2π)Deik(xb−xa)

ih

E − h2k2/2M + iη. (9.4)

760

9.1 Free Particle in D Dimensions 761

The momentum integral can now be done as follows. The exponential functionexp (ikR) is written as exp (ikR cosϑ), where R is the distance vector xb − xa andϑ the angle between k and R. Then we use formula (8.100) with the coefficients(8.101) and the hyperspherical harmonics Ylm(x) of Eq. (8.126) and expand

eikR =∑

l=0

al(ikR)∑

m

Ylm(k)Y ∗lm(R). (9.5)

The integral over k follows now directly from the decomposition into size and di-rection dDk = dkdk, and the orthogonality property (8.115) of the hypersphericalharmonics, according to which

∫

dkYlm(k) = δl0δm0

√

SD, (9.6)

with SD of Eq. (1.558). Since Y00(x) = 1/√SD, we obtain

(xb|xa)E = − 2Mi

(2π)D

∫ ∞

0dkkD−1 1

k2 + κ2a0(ikR), (9.7)

where

κ ≡√

−2ME/h2, (9.8)

as in (1.346). Inserting a0(ikR) from (8.101),

a0(ikR) = (2π)D/2JD/2−1(kR)/(kR)D/2−1, (9.9)

we find

(xb|xa)E = − 2Mi

(2π)D/2R1−D/2

∫ ∞

0dkkD/2 1

k2 + κ2JD/2−1(kR). (9.10)

The integral∫ ∞

0dk

kν+1

(k2 + a2)µ+1Jν(kb) =

aν−µbµ

2µΓ(µ+ 1)Kν−µ(ab) (9.11)

yields once more the fixed-energy amplitude (1.347).In two dimensions, the amplitude (1.347) becomes [recall (1.350)]

(xb|xa)E = −iMπh

K0(κ|xb − xa|). (9.12)

It can be decomposed into partial waves by inserting

|xb − xa| =√

r2b + r2a − 2rarb cos(ϕb − ϕa). (9.13)

Then a well-known addition theorem for Bessel functions yields the expansion

K0(κ|xb − xa|) =∞∑

m=−∞

Im(κr<)Km(κr>)eim(ϕb−ϕa), (9.14)

762 9 Wave Functions

where r< and r> are the smaller and larger values of ra and rb, respectively. Hencethe fixed-energy amplitude turns into

(xb|xa)E = −2iM

h

∑

m

Im(κr<)Km(κr>)1

2πeim(ϕb−ϕa). (9.15)

This is an analytic function in the complex E-plane. The parameter κ is real forE < 0. For E > 0, the square root (9.8) allows for two imaginary solutions,κ± ≡ e∓iπ/2k ≡ e∓iπ/2

√2ME/h, so that the amplitude has a right-hand cut. Its

discontinuity specifies the continuum of free-particle states. On top of the cut, weuse the analytic continuation formulas (valid for −π/2 < arg z ≤ π)1

Iµ(e−iπ/2z) = e−iπµ/2Jµ(z),

Kµ(e−iπ/2z) = i

π

2eiπµ/2H(1)

µ (z) , (9.16)

to find the fixed-energy amplitude above the cut

(xb|xa)E+iη =πM

h

∑

m

Jm(kr<)H(1)m (kr>)

1


The reflection properties2

H(1)µ (eiπz) = −H(2)

−µ(z) ≡ −e−iπµH(2)µ (z),

Jµ(eiπz) = eiπµJµ(z) (9.18)

yield the amplitude below the cut

(xb|xa)E−iη = −πMh

∑

m

Jm(kr<)H(2)m (kr>)

1


The discontinuity across the cut follows from the relation

Jµ(z) =1

2

[

H(1)µ (z) +H(2)

µ (z)]

(9.20)

and reads

disc (xb|xa)E =2πM

h

∞∑

m=−∞

Jm(krb)Jm(kra)1

2πeim(ϕb−ϕb). (9.21)

According to (1.330), the integral over the discontinuity yields the completenessrelation

∫ ∞

−∞

dE

2πhdisc (xb|xa)E

=∫ ∞

−∞

dE

2πh

2πM

h

∞∑

m=−∞

Jm(krb)Jm(kra)1

2πeim(ϕb−ϕb) = δ(xb − xa). (9.22)

1I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 8.406.1 and 8.407.1.2ibid., Formulas 8.476.1 and 8.476.8.

H. Kleinert, PATH INTEGRALS

9.2 Harmonic Oscillator in D Dimensions 763

After replacing the energy integral by a k-integral,∫ ∞

−∞

dE

2πh

2πM

h=∫ ∞

0dkk, (9.23)

we can also write∫ ∞

−∞

dE

2πhdisc (xb|xa)E =

∫ d2k

(2π)2eik(xb−xa) =

1

2π

∫ ∞

0dkkJ0(k|xb − xa|)

=∞∑

m=−∞

∫ ∞

0dkkJm(krb)Jm(kra)

1

2πeim(ϕb−ϕa)

=1√rbra

δ(rb − ra)δ(ϕb − ϕa). (9.24)

The last two lines exhibit the well-known completeness relation of the radial wavefunctions of a free particle.3

9.2 Harmonic Oscillator in D Dimensions

The wave functions of the one-dimensional harmonic oscillator have already beenderived in Section 2.6 from a spectral decomposition of the time evolution ampli-tude. This was possible with the help of Mehler’s formula. In D dimensions, thefixed-energy amplitude is the best starting point for determining the wave functions.We take the radial propagator (8.142) obtained from the angular momentum decom-position (8.141) or, for the sake of greater generality, the radial amplitude (8.144)with an additional centrifugal barrier, continue it to imaginary time τ = it, and goover to its Laplace transform

(rb|ra)E,l = −i√

Mω/h√

Mωrbra/h∫ ∞

τadτb e

E(τb−τa)/h1

sin[ω(τb − τa)]

×e− 1h

Mω2

coth[ω(τb−τa)](r2b+r2a)Iµ

(

Mωrbrah sinh[ω(τb − τa)]

)

. (9.25)

To evaluate the τ -integral we make use of a standard integral formula for Besselfunctions4∫ ∞

0dx [coth(x/2)]2ν e−β cosh xJµ(α sinh x)

=Γ((1 + µ)/2− ν)

αΓ(µ+ 1)Wν,µ/2

(

√

α2 + β2 + β)

M−ν,µ/2

(

√

α2 + β2 − β)

,(9.26)

where Wν,µ/2(z), M−ν,µ/2(z) are the Whittaker functions. The formula is valid for

Re β > |Reα|, Re (µ/2− ν) > −1

2.

3Compare with Eqs. (3.112) and (3.139) in J.D. Jackson, Classical Electrodynamics , John Wiley& Sons, New York, 1975.

4I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 6.669.1.


By a change of variables√

α2 + β2 ± β = tαb,a,

and

sinh x = (sinh y)−1, cosh x = coth y, coth(x/2) = ey, coth x = cosh y,

withdx = dy/ sinh y,

Eq. (9.26) goes over into

∫ ∞

0

dy

sinh ye2νy exp [− 1

2t(αb − αa) coth y]Jµ

(

t

√αbαa

sinh y

)

=Γ ((1 + µ)/2− ν)

t√αbαaΓ(µ+ 1)

Wν,µ/2(tαb)M−ν,µ/2(−tαa). (9.27)

Using the identityM−ν,µ/2(z) ≡ e−i(µ+1)π/2Mν,µ/2(−z) (9.28)

and changing the sign of αa in (9.27), this can be turned into

∫ ∞

0

dy

sinh ye2νy exp [− 1

2 t(αb + αa) coth y] Iµ

(

t√αbαa

sinh y

)

=Γ ((1 + µ)/2− ν)

t√αbαaΓ(µ+ 1)

Wν,µ/2(tαb)Mν,µ/2(tαa), (9.29)

with the range of validity

αb > αa > 0, Re [(1 + µ)/2− ν] > 0,

Re t > 0, |arg t| < π. (9.30)

Setting

y = ω(tb − ta), αb =M

hωr2b , αa =

Mω

hr2a, ν = E/2ωh (9.31)

in (9.29) brings the radial amplitude (9.25) to the form (valid for rb > ra)

(rb|ra)E,l = −i 1ω

1√rbra

Γ((1 + µ)/2− ν)

Γ(µ+ 1)Wν,µ/2

(

Mω

hr2b

)

Mν,µ/2

(

Mω

hr2a

)

. (9.32)

The Gamma function has poles at

ν = νr ≡ (1 + µ)/2 + nr (9.33)

for integer values of the so-called radial quantum number of the system nr =0, 1, 2, . . . . The poles have the form

Γ ((1 + µ)/2− ν)ν∼νr∼ −(−1)nr

nr!

1

ν − νr. (9.34)



Inserting here the particular value of the parameter µ for the D-dimensional oscilla-tor which is µ = D/2+ l− 1, and remembering that ν = E/2ωh, we find the energyspectrum

E = hω (2nr + l +D/2) . (9.35)

The principal quantum number is defined by

n ≡ 2nr + l (9.36)

and the energy depends on it as follows:

En = hω(n+D/2). (9.37)

For a fixed principal quantum number n = 2nr + l, the angular momentum runsthrough l = 0, 2, . . . , n for even, and l = 1, 3, . . . , n for odd n. There are dn = (n +D−1)!/(D−1)!n! degenerate levels. From the residues 1/(ν − νr) ∼ 2hω/(E−En),we extract the product of radial wave functions at given nr, l:

Rnrl(rb)Rnrl(ra) =1√rbra

2(−1)nr

Γ(µ+ 1)nr!(9.38)

×W(1+µ)/2+nr ,µ

2(Mωrb

2/h)M(1+µ)/2+nr ,µ

2(Mωra

2/h).

It is now convenient to express the Whittaker functions in terms of the confluenthypergeometric or Kummer functions:5

W(1+µ)/2+nr ,µ

2(z) = e−z/2z(1+µ)/2U(−nr , 1 + µ, z), (9.39)

M(1+µ)/2+nr ,µ

2(z) = e−z/2z(1+µ)/2M(−nr , 1 + µ, z). (9.40)

The latter equation follows from the relation

M−(1+µ)/2−nr ,µ

2(z) = e−z/2z(1+µ)/2M(1 + µ+ nr, 1 + µ, z), (9.41)

after replacing nr → −nr − µ− 1.For completeness, we also mention the identity

M(a, b, z) = ezM(b− a, b,−z), (9.42)

so thatM(1 + µ+ nr, 1 + µ, z) = ezM(−nr , 1 + µ,−z). (9.43)

This permits us to rewrite (9.41) as

M−(1+µ)/2−nr ,µ

2(z) = ez/2z(1+µ)/2M(−nr , 1 + µ,−z), (9.44)

which turns into (9.40) by using (9.28) and appropriately changing the indices.

5I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 9.220.2.


The Kummer function M(a, b, z) has the power series

M(a, b, z) ≡ 1F1(a; b; z) = 1 +a

bz +

a(a + 1)

b(b+ 1)

z

z!+ . . . , (9.45)

showing that M(1+µ)/2+nr ,µ

2(Mωra

2/2h) is an exponential e−Mωr2a/h times a poly-nomial in ra of order 2nr. A similar expression is obtained for the other factorW(1+µ)/2+nr ,

µ

2(Mωrb

2/h) of Eq. (9.39). Indeed, the Kummer function U(a, b, z) is

related to M(a, b, z) by6

U(a, b, z) =π

sin πb

[

M(a, b, z)

Γ(1 + a− b)Γ(b)− z1−bM(1 + a− b, 2− b, z)

Γ(a)Γ(2− b)

]

. (9.46)

Since a = −nr with integer nr and 1/Γ(a) = 0, we see that only the first term inthe brackets is present. Then the identity

Γ(−µ)Γ(1 + µ) = π/ sin[π(1 + µ)]

leads to the relation

U(−nr, 1 + µ, z) =Γ(−µ)

Γ(−nr − µ)M(−nr, 1 + µ, z), (9.47)

which is a polynomial in z of order nr. Thus we have the useful formula

W(1+µ)/2+nr ,µ

2(zb)M(1+µ)/2+nr ,

µ

2(za) =

Γ(−µ)Γ(−nr − µ)

e−(zb+za)/2

×(zbza)(1+µ)/2M(−nr , 1 + µ, zb)M(−nr , 1 + µ, za). (9.48)

We can therefore re-express Eq. (9.38) as

Rnrl(rb)Rnrl(ra) =

√

Mω

h

2(−)nrΓ(−µ)Γ(−nr − µ)Γ(1 + µ) nr!

×e−Mω(r2b+r2a)/2h (Mωrbra/h)

1/2+µ (9.49)

×M(−nr , 1 + µ,Mωr2b/h)M(−nr, 1 + µ,Mωr2a/h).

We now insert(−)nrΓ(−µ)Γ(−nr − µ)

=Γ(nr + 1 + µ)

Γ(1 + µ), (9.50)

setting µ = D/2 + l − 1, and identify the wave functions as

Rnrl(r) = Cnrl (Mω/h)1/4(Mωr2/h)l/2+(D−1)/4

×e−Mωr2/2hM(−nr , l +D/2,Mωr2/h), (9.51)

6M. Abramowitz and I. Stegun, Handbook of Mathematical Functions , Dover, New York, 1965,Formula 13.1.3.



with the normalization factor

Cnrl =

√2

Γ(1 + µ)

√

(nr + µ)!

nr!. (9.52)

By introducing the Laguerre polynomials 7

Lµn(z) ≡

(n + µ)!

n!µ!M(−n, µ + 1, z), (9.53)

and using the integral formula8

∫ ∞

0dze−zzµLµ

n(z)Lµn′(z) = δnn′

(n+ µ)!

n!, (9.54)

we find that the radial wave functions satisfy the orthonormality relation∫ ∞

0drRnrl(r)Rn′

rl(r) = δnrn′r. (9.55)

The radial imaginary-time evolution amplitude has now the spectral representa-tion

(rbτb|raτa)l =∞∑

nr=0

Rnrl(rb)Rnrl(ra)e−En(τb−τa)/h, (9.56)

with the energies

En = hω(n+D/2) = hω (2nr + l +D/2) . (9.57)

The full causal propagator is given, as in (8.91), by

(xbτb|xaτa) =1

(rbra)(D−1)/2

∞∑

l=0

(rbτb|raτa)l∑

m

Ylm(xb)Y∗lm(xa). (9.58)

From this, we extract the wave functions

ψnrlm(x) =1

r(D−1)/2Rnrl(r)Ylm(x). (9.59)

They have the threshold behavior rl near the origin.The one-dimensional oscillator may be viewed as a special case of these formulas.

For D = 1, the partial wave expansion amounts to a separation into even and oddwave functions. There are two “spherical harmonics”,

Ye,o(x) =1√2(Θ(x)±Θ(−x)), (9.60)

7I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.970 (our definition differs from that inL.D. Landau and E.M. Lifshitz, Quantum Mechanics , Pergamon, London, 1965, Eq. (d.13). Therelation is Lµ

n= (−)µ/(n + µ)!LL.L.

n+µ

µ).8ibid., Formula 7.414.3.


and the amplitude has the decomposition

(xbτb|xaτa) = (rbτb|raτa)eYe(xb)Ye(xa) + (rbτb|raτa)oYo(xb)Yo(xa), (9.61)

with the “radial” amplitudes

(rbτb|raτa)e,o = (xbτb|xaτa)± (−xbτb|xaτa). (9.62)

These are known from Eq. (2.177) to be

(rbτb|raτa)e,o =1√2π

√

√

√

√

Mω/h

sinh[ω(τb − τa)](9.63)

× exp[

−Mω

2h(r2b + r2a) cotω(τb − τa)

]

2

{

coshsinh

}

(Mωrbra/h) .

The two cases coincide with the integrand of (9.25) for l = 0 and 1, respectively,since µ = l +D/2− 1 takes the values ±1/2 and

√zI∓ 1

2(z) =

2

2π

{

cosh z ,sinh z .

(9.64)

The associated energy spectrum (9.35) is

E =

{

hω(2nr +12) even,

hω(2nr +32) odd,

(9.65)

with the radial quantum number nr = 0, 1, 2, . . . . The two cases follow the singleformula

E = hω(n+ 12), (9.66)

where the principal quantum number n = 0, 1, 2, . . . is related to nr by n = 2nr andn = 2nr + 1, respectively. The radial wave functions (9.51) become

Rnr,e(r) = (Mω/h)1/4

√

√

√

√

2Γ(nr +12)

πnr!M(−nr, 1

2 ,Mωr2/h), (9.67)

Rnr,o(r) = (Mω/h)1/4

√

√

√

√

2Γ(nr +32)

(π/4)nr!

√

Mωr2

hM(−nr, 3

2 ,Mωr2/h). (9.68)

The special Kummer functions appearing here are Hermite polynomials

M(−n, 12 , x

2) =n!

(2n)!(−)nH2n(x), (9.69)

M(−n, 32 , x

2) =n!

(2n+ 1)!(−)nH2n+1(x)/2

√x. (9.70)


9.3 Free Particle from ω → 0 -Limit of Oscillator 769

Using the identityΓ(z)Γ(z + 1

2) = (2π)1/22−2z+1/2Γ(2z), (9.71)

we obtain in either case the radial wave functions [to be compared with the one-dimensional wave functions (2.302)]

Rn(r) = Nn

√2λ−1/2

ω e−r2/2λ2ωHn(r/λω), n = 0, 1, 2, . . . (9.72)

with

λω ≡√

h

Mω, Nn =

1√

2nn!√π. (9.73)

This formula holds for both even and odd wave functions with nr = 2n and nr =2n+ 1, respectively. It is easy to check that they possess the correct normalization∫∞0 drR2

n(r) = 1. Note that the “spherical harmonics” (9.60) remove a factor√2

in (9.72), but compensate for this by extending the x > 0 integration to the entirex-axis by the “one-dimensional angular integration”.

9.3 Free Particle from ω → 0 -Limit of Oscillator

The results obtained for theD-dimensional harmonic oscillator in the last section canbe used to find the amplitude and wave functions of a free particle in D dimensionsin radial coordinates. This is done by taking the limit ω → 0 at fixed energy E.In the amplitude (9.32) with Wν,µ/2(z),Mν,µ/2(z) substituted according to (9.39),we rewrite nr as (E/ωh− l − 1) /2 and go to the limit ω → 0 at a fixed energyE. Replacing Mωr2/h by k2r2/2nr≡ z/nr (where z = k2r2/2, and using E =p2/2M=h2k2/2M), we apply the limiting formulas9

limnr→∞

{Γ(1− nr − b)U (−a, b,∓z/nr)}

= z−12(b−1)

{

2Kb−1(2√z)

−iπeiπbH(1)b−1(2

√z) (Im z > 0),

(9.74)

limnr→∞

M (−a, b,∓z/nr) /Γ(b) = z−12(b−1)

{

Ib−1(2√z)

Jb−1(2√z),

(9.75)

and obtain the radial wave functions directly from (9.51) and (9.75):

Rnrl(r)nr−−−→∞

−−−→ Cnrl(Mωr2/h)(µ/2+1/2)(

k2r2/2)−µ/2

Γ(1 + µ)Jµ(kr), (9.76)

where

Cnrl

nr−−−→∞

−−−→√2(

E

2hω

)µ/2 1

Γ(1 + µ). (9.77)

HenceRnrl(r)

nr−−−→∞

−−−→ r1/2√

Mω/h√2Jµ(kr). (9.78)

9M. Abramowitz and I. Stegun, op. cit., Formulas 13.3.1–13.3.4.


Inserting these wave functions into the radial time evolution amplitude

(rbτb|raτa)l =∑

nr

Rnrl(rb)Rnrl(rb)e−En(τb−τa)/h, (9.79)

and replacing the sum over nr by the integral∫∞0 dk hk/Mω [in accordance with

the nr → ∞ limit of Enr= ωh(2nr + l +D/2) → h2k2/2M ], we obtain the spectral

representation of the free-particle propagator

(rbτb|raτa)µ =√rbra

∫ ∞

0dk kJµ(krb)Jµ(kra)e

− hk2

2M(τb−τa). (9.80)

For comparison, we derive the same results directly from the initial spectralrepresentation (9.2) in one dimension

(xbτb|xaτa) =∫ ∞

−∞

dk

2πexp

[

ik(xb − xa)−hk2

2M(τb − τa)

]

. (9.81)

Its “angular decomposition” is a decomposition with respect to even and odd wavefunctions

(rbτb|raτa)e,o =∫ ∞

0

dk

π[cos k(rb − ra)± cos k(rb + ra)]e

− hk2

2M(τb−τa)

= 2∫ ∞

0

dk

π

{

cos krb cos krasin krb sin kra

}

e−hk2

2M(τb−τa). (9.82)

In D dimensions we use the expansion (8.100) for eikx to calculate the amplitude inthe radial form

(xbτb|xaτa) =∫

dDk

(2π)Deik(xb−xa)e−

hk2

2M(τb−τa)

=1

(2π)D/2

∫ ∞

0dkk2ν

1

(kR)νJν(kR)e

− hk2

2M(τb−τa), (9.83)

with ν ≡ D/2 − 1. With the help of the addition theorem for Bessel functions10

(8.188) we rewrite

1

(kR)νJν(kR) =

2νΓ(ν)

(k2rbra)ν

∞∑

l=0

(ν + l)Jν+l(krb)Jν+l(kra)C(ν)l (∆ϑ) (9.84)

and expand further according to

1

(kR)νJν(kR) =

(2π)D/2

(k2rbra)ν

∞∑

l=0

Jν+l(krb)Jν+l(kra)∑

m

Ylm(xb)Y∗lm(xa), (9.85)

to obtain the radial amplitude

(rbτb|raτa)l =√rbra

∫ ∞

0dkkJν+l(krb)Jν+l(kra)e

− hk2

2M(τb−τa), (9.86)

10I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.532.


9.4 Charged Particle in Uniform Magnetic Field 771

just as in (9.80).For D = 1, this reduces to (9.82) using the particular Bessel functions

√zJ∓1/2(z) =

2√2π

{

cos zsin z

}

. (9.87)

9.4 Charged Particle in Uniform Magnetic Field

Let us also find the wave functions of a charged particle in a magnetic field. Theamplitude was calculated in Section 2.18. Again we work with the imaginary-timeversion. Factorizing out the free motion along the direction of the magnetic field,we write

(xbτb|xaτa) = (zbτb|zaτa)(x⊥b τb|x⊥

a τa), (9.88)

with

(zbτb|zaτa) =1

√

2πh(τb − τa)/Mexp

{

−M2h

(zb − za)2

τb − τa

}

, (9.89)

and have for the amplitude in the transverse direction

(x⊥b τb|x⊥

a τa) =M

2πh(τb − τa)

ω(τb − τa)/2

sinh [ω(τb − τa)/2]exp

[

−A⊥l /h

]

, (9.90)

with the classical transverse action

A⊥l =

Mω

2

{

12 coth [ω(τb − τa)/2] (x

⊥b − x⊥

a )2 + x⊥

a × x⊥b

}

. (9.91)

This result is valid if the vector potential is chosen as

A =1

2B× x. (9.92)

In the other gauge withA = (0, Bx, 0), (9.93)

there is an extra surface term, and A⊥cl is replaced by

A⊥cl = A⊥

cl +Mω

2(xbyb − xaya). (9.94)

The calculation of the wave functions is quite different in these two gauges. In thegauge (9.93) we merely recall the expressions (2.663) and (2.665) and write downthe integral representation

(x⊥b τb|x⊥

a τa) =∫

dpy2πh

eipy(yb−ya)/h(xbτb|xaτa)x0=py/Mω, (9.95)

with the oscillator amplitude in the x-direction

(xbτb|xaτa)x0=

√

Mω

2πh sinh[ω(τb − τa)]exp

(

−1

hAos

cl

)

, (9.96)


and the classical oscillator action centered around x0

Aoscl =

Mω

2 sinh[ω(τb − τa)]{[(xb − x0)

2 + (xa − x0)2] cosh[ω(τb − τb)]

− 2(xb − x0)(xa − x0)}. (9.97)

The spectral representation of the amplitude (9.96) is then

(xaτb|xaτa)x0=

∞∑

n=0

ψn(xb − x0)ψn(xa − x0)e−(n+ 1

2)ω(τb−τa), (9.98)

where ψn(x) are the oscillator wave functions (2.302). This leads to the spectralrepresentation of the full amplitude (9.88)

(xbτb|xaτa) =∫ dpz

2πh

∫ dpy2πh

eipz(zb−za)/h (9.99)

×∞∑

n=0

ψn(xb − py/Mω)ψn(xa − py/Mω)e−(n+ 12)ω(τb−τa).

The combination of a sum and two integrals exhibits the complete set of wavefunctions of a particle in a uniform magnetic field. Note that the energy

En = (n+ 12)hω (9.100)

is highly degenerate; it does not depend on py.In the gauge A = 1

2B × x, the spectral decomposition looks quite different.

To derive it, the transverse Euclidean action is written down in radial coordinates[compare Eq. (2.669)] as

A⊥cl =

M

2

{

ω

2coth [ω(τb − τa)/2]

[

rb2 + ra

2 − 2rbra cos(ϕb − ϕa)]

− iωrbra sin(ϕb − ϕa)}

. (9.101)

This can be rearranged to

A⊥cl =

M

2

ω

2coth [ω(τb − τa)/2] (rb

2 + ra2)

−M2

ω

sinh [ω(τb − τa)/2]cos [ϕb − ϕa − iω(τb − τa)/2] . (9.102)

We now expand e−A⊥

cl/h into a series of Bessel functions using (8.5)

e−A⊥

cl/h = exp

{

−M2h

ω

2coth [ω(τb − τa)/2] (rb

2 + ra2)}

(9.103)

×∞∑

m=−∞

Im

(

Mω

2h

rbrasinh [ω(τb − τa)/2]

)

emω(τb−τa)/2eim(ϕb−ϕa).



The fluctuation factor is the same as before. Hence we obtain the angular decom-position of the transverse amplitude

(x⊥b τb|x⊥

a τa) =1√rbra

∑

m

(rbτb|raτa)m1

2πeim(ϕb−ϕa), (9.104)

where

(rbτb|raτa)m =√rbra

Mω

2hη

η

sinh ηexp

[

−M2h

ω

2coth η(r2b + r2a)

]

Im

(

Mωrbra2h sinh η

)

emη,

(9.105)

with

η ≡ ω(τb − τa)/2. (9.106)

To find the spectral representation we go to the fixed-energy amplitude

(rb|ra)m,E = −i∫ ∞

τadτbe

E(τb−τa)/h (rbτb|raτa)m (9.107)

= −i√rbraM

h

∫ ∞

0dη e2νη

1

sinhηe−(Mω/4h) coth η(r2

b+r2a)Im

(

Mωrbra2hsinhη

)

.

The integral is done with the help of formula (9.29) and yields

(rb|ra)m,E = −i√rbraM

h

Γ(

12− ν + |m|

2

)

(Mω/2h)rbraΓ(|m|+ 1)

×Wν,|m|/2

(

Mω

2hr2b

)

Mν,|m|/2

(

Mω

2hr2a

)

, (9.108)

with

ν ≡ E

ωh+m

2. (9.109)

The Gamma function Γ(1/2− ν − |m|/2) has poles at

ν = νr ≡ nr +1

2+

|m|2

(9.110)

of the form

Γ(1/2− ν − |m|/2) ≈ − 1

nr!

(−1)nr

ν − νr∼ −(−1)nr

nr!

ωh

E − Enrm. (9.111)

The poles lie at the energies

Enrm = hω

(

nr +1

2+

|m|2

− m

2

)

. (9.112)


These are the well-known Landau levels of a particle in a uniform magnetic field.The Whittaker functions at the poles are (for m > 0)

M−ν,m/2(z) = ez/2z1+m

2 M(−nr, 1 +m,−z), (9.113)

Wν,m/2(z) = e−z/2z1+m

2 (−)nr(nr +m)!

m!M(−nr, 1 +m, z). (9.114)

The fixed-energy amplitude near the poles is therefore

(rb|ra)m,E ∼ ih

E −EnrmRnrm(rb)Rnrm(ra), (9.115)

with the radial wave functions11

Rnrm(r) =√r(

Mω

h

)1/2√

(nr + |m|)!nr!

1

|m|! (9.116)

× exp(

−Mω

4hr2)(

Mω

2hr2)|m|/2

M(

−nr, 1 + |m|, Mω

2hr2)

.

Using Eq. (9.53), they can be expressed in terms of Laguerre polynomials Lαn(z):

Rnrm =√r(

Mω

h

)1/2√

nr!

(nr + |m|)! exp(

−Mω

4hr2)

×(

Mω

2hr2)|m|/2

L|m|nr

(

Mω

2hr2)

. (9.117)

The integral (9.54) ensures the orthonormality of the radial wave functions

∫ ∞

0drRnrm(r)Rn′

rm(r) = δnrn′r. (9.118)

A Laplace transformation of the fixed-energy amplitude (9.108) gives, via the residuetheorem, the spectral representation of the radial time evolution amplitude

(rbτb|raτa) =∑

nrm

Rnrm(rb)Rnrm(ra)e−Enrm(τb−τa)/h, (9.119)

with the energies (9.112). The full wave functions in the transverse subspace are, ofcourse,

ψnrm(x) =1√rRnrm(r)

eimϕ

√2π. (9.120)

Comparing the energies (9.112) with (9.100), we identify the principal quantumnumber n as

n ≡ nr +|m|2

− m

2. (9.121)

11Compare with L.D. Landau and E.M. Lifshitz, Quantum Mechanics , Pergamon, London, 1965,p. 427.



Note that the infinite degeneracy of the energy levels observed in (9.100) with respectto py is now present with respect tom. This energy does not depend onm form ≥ 0.The somewhat awkward m-dependence of the energy can be avoided by introducing,instead of m, another quantum number n′ related to n,m by

m = n′ − n. (9.122)

The states are then labeled by n, n′ with both n and n′ taking the values 0, 1, 2, 3, . . . .For n′ < n, one has n′ = nr and m = n′ − n < 0, whereas for n′ ≥ n one has n = nr

and m = n′ − n ≥ 0. There exists a natural way of generating the wave functionsψnrm(x) such that they appear immediately with the quantum numbers n, n′. Forthis we introduce the Landau radius

a =

√

2h

Mω=

√

2hc

eB(9.123)

as a length parameter and define the dimensionless transverse coordinates

z = (x+ iy)/√2a, z∗ = (x− iy)/

√2a. (9.124)

It is then possible to prove that the ψnrm’s coincide with the wave functions

ψn,n′(z, z∗) = Nn,n′ez∗z

(

− 1√2∂z∗

)n (

− 1√2∂z

)n′

e−2z∗z. (9.125)

The normalization constants are obtained by observing that the differential opera-tors

ez∗z

(

− 1√2∂z∗

)

e−z∗z =1√2(−∂z∗ + z),

ez∗z

(

− 1√2∂z

)

e−z∗z =1√2(−∂z + z∗) (9.126)

behave algebraically like two independent creation operators

a† =1√2(−∂z∗ + z),

b† =1√2(−∂z + z∗), (9.127)

whose conjugate annihilation operators are

a =1√2(∂z + z∗),

b =1√2(∂z∗ + z). (9.128)


The ground state wave function annihilated by these is

ψ0,0(z, z∗) = 〈z, z∗|0〉 ∝ e−z∗z. (9.129)

We can therefore write the complete set of wave functions as

ψn,n′(z, z∗) = Nnn′ a†nb†n′

ψ0,0(z, z∗). (9.130)

Using the fact that a†∗ = b†, b†∗ = a†, and that partial integrations turn b†, a† intoa, b, respectively, the normalization integral can be rewritten as

∫

dx dy ψn1,n′1(z, z∗)ψn2,n′

2(z, z∗)

= Nn1n′1Nn2n′

2

∫

dxdy[

(a†)n1(b†)n′1e−z∗z

] [

(a†)n2(b†)n′2e−z∗z

]

= Nn1n′1Nn2n′

2

∫

dxdye−2z∗z(

an1bn′1a†n2b†n

′2

)

. (9.131)

Here the commutation relations between a†, b†, a, b serve to reduce the parenthesesin the last line to

n1!n2! δn1n′1δn2n′

2. (9.132)

The trivial integral∫

dxdy e−2z∗z = π∫

dr2 e−r2/a2 = πa2 (9.133)

shows that the normalization constants are

Nn,n′ =1√

πa2n!n′!. (9.134)

Let us prove the equality of ψnrm and ψn,n′ up to a possible overall phase. For this wefirst observe that z, ∂z∗ and z∗, ∂z carry phase factors eiϕ and e−iϕ, respectively, sothat the two wave functions have obviously the azimuthal quantum number m = n−n′. Second, we make sure that the energies coincide by considering the Schrodingerequation corresponding to the action (2.643)

1

2M

(

−ih∇− e

cA

)2

ψ = Eψ. (9.135)

In the gauge whereA = (0, Bx, 0),

it reads

− h2

2M

[

∂x2 + (∂y − i

eB

cx)2 + ∂z

2]

ψ = Eψ, (9.136)

and the wave functions can be taken from Eq. (9.99). In the gauge where

A = (−By/2, Bx/2, 0) , (9.137)



on the other hand, the Schrodinger equation becomes, in cylindrical coordinates,

[

− h2

2M

(

∂2r +1

r∂r +

1

r2∂2ϕ + ∂2z

)

− iehB

2Mc∂ϕ +

e2B2

8Mc2r2]

ψ(r, z, ϕ)

= Eψ(r, z, ϕ). (9.138)

Employing a reduced radial coordinate ρ = r/a and factorizing out a plane wave inthe z-direction, eipzz/h, this takes the form

[

∂2ρ +1

ρ∂ρ +

a2

h2(2ME − p2z)− ρ2 − 2i∂ϕ − 1

ρ2∂2ϕ

]

ψ(r, ϕ) = 0. (9.139)

The solutions are

ψnrm(r, ϕ) ∝ eimϕe−ρ2/2ρ|m|/2M(

−nr, |m|+ 1

2, ρ)

, (9.140)

where the confluent hypergeometric functions M(

−nr, |m|+ 12, ρ)

are polynomialsfor integer values of the radial quantum number

nr = n+1

2m− 1

2|m| − 1

2, (9.141)

as in (9.116). The energy is related to the principal quantum number by

n+1

2≡ a2

h2

(

2ME − p2z)

. (9.142)

Since2Ma2

h2=

1

hω, (9.143)

the energy is

E =(

n +1

2

)

hω +p2z2M

. (9.144)

We now observe that the Schrodinger equation (9.139) can be expressed in terms ofthe creation and annihilation operators (9.127), (9.128) as

4

[

−(a†a + 1/2) +1

hω

(

E − p2z2M

)]

ψ(z, z∗) = 0. (9.145)

This proves that the algebraically constructed wave functions ψn,n′ in (9.130) coin-cide with the wave functions ψnrm of (9.116) and (9.140), up to an irrelevant phase.Note that the energy depends only on the number of a-quanta; it is independent ofthe number of b-quanta.


9.5 Dirac δ -Function Potential

For a particle in a Dirac δ-function potential, the fixed-energy amplitudes (xb|xa)Ecan be calculated by performing a perturbation expansion around a free-particle am-plitude and summing it up exactly. For any time-independent potential V (x), in ad-dition to a harmonic potential Mω2x2/2, the perturbation expansion in Eq. (3.478)can be Laplace-transformed in the imaginary time via (9.3) to find

(xb|xa)E = (xb|xa)ω,E − i

h

∫

dDx1(xb|x1)ω,EV (x1)(x1|xa)ω,E

+ − 1

h2

∫

dDx2

∫

dDx1(xb|x2)ω,EV (x2)(x2|x1)ω,EV (x1)(x1|xa)ω,E

+ . . . . (9.146)

If the potential is a Dirac δ-function centered around X,

V (x) = g δ(D)(x−X), g ≡ h2

Ml2−D, (9.147)

this series simplifies to

(xb|xa)E=(xb|xa)ω,E−ig

hg(xb|X)ω,E(X|xa)ω,E−

g2

h2(xb|X)ω,E(X|X)ω,E(X|xa)ω,E+. . . ,

(9.148)

and can be summed up to

(xb|xa)E = (xb|xa)ω,E − ig

h

(xb|X)ω,E(X|xa)ω,E

1 + ig

h(X|X)ω,E

. (9.149)

This is, incidentally, true if a δ-function potential is added to an arbitrary solvablefixed-energy amplitude, not just the harmonic one.

If the δ-function is the only potential, we use formula (9.149) with ω = 0, sothat (xb|xa)0,E reduces to the fixed-energy amplitude (9.12) of a free particle, andobtain directly

(xb|xa)E = −2iM

h

κD−2

(2π)D/2

KD/2−1(κR)

(κR)D/2−1

− ig

h

i2M

h

κD−2

(2π)D/2

KD/2−1(κRb)

(κRb)D/2−1× i

2M

h

κD−2

(2π)D/2

KD/2−1(κRa)

(κRa)D/2−1

1− g

h

2M

h

κD−2

πD/2

KD/2−1(κδ)

(κδ)D/2−1

, (9.150)

where R ≡ |xb − xa| and Ra,b ≡ |xa,b − X|, and δ is an infinitesimal distanceregularizing a possible singularity at zero-distance. In D = 1 dimension, this reducesto

(xb|xa)E = −iMh

1

κe−κR + i

M

hκeκ(Rb+Ra)

1

lκ+ 1, (9.151)


9.5 Dirac δ-Function Potential 779

For an attractive potential with l < 0, the second term can be written as

−i 1

κl2h

E + h2/2Ml2eκ(Rb+Ra), (9.152)

exhibiting a pole at the bound-state energy EB = −h/2Ml2. In its neighborhood,the pole contribution reads

2

le−(Rb+Ra)/l

ih

E + h2/2Ml2. (9.153)

This has precisely the spectral form (1.325) with the normalized bound-state wavefunction

ψB(x) =

√

2

le−|x−X|/l. (9.154)

In D = 3 dimensions, the amplitude (9.150) becomes

(xb|xa)E = −iMh

1

2πRe−κR + i

M

h

eκRb

2πRb

eκRa

2πRa

1

1/l + e−κδ/2πδ. (9.155)

In the limit δ → 0, the denominator requires renormalization. We introduce arenormalized coupling length scale

1

lr≡ 1 +

1

2πδ, (9.156)

and rewrite the last factor in (9.155) as

1

1/lr − κ/2π. (9.157)

For lr < 0, this has a pole at the bound-state energy EB = −4π2h2/2Ml2R of theform

−lrEB1

E −EB

. (9.158)

The total pole term in (9.155) can therefore be written as

ψB(xb)ψ∗B(xa)

ih

E − EB

, (9.159)

with κB =√2MEB/h = 2π/lr and the normalized bound-state wave functions

ψB(x) =

(

κ2B4π2

)1/4e−κB|x−X|

r. (9.160)


In D = 2 dimensions, the situation is more subtle. It is useful to consider theamplitude (9.150) in D = 2 + ǫ dimensions where one has

(xb|xa)E = −iMh

1

π

Kǫ/2(κR)

(2πκR)ǫ/2

+ iM2

h2π2

1

(2πκRb)ǫ/21

(2πκRa)ǫ/2Kǫ/2(κRb)Kǫ/2(κRa)

h

g+M

hπ

1

(2πκδ)ǫ/2Kǫ/2(κǫ)

. (9.161)

Inserting here Kǫ/2(κδ) ≈ (1/2)Γ(ǫ/2)(κδ/2)−ǫ/2, the denominator becomes

h

g+

M

2hπ

Γ(ǫ/2)

(πκδ)ǫ/2≈ h

g+

M

2hπ

2

ǫ

[

1− ǫ

2log(πκδ)

]

. (9.162)

Here we introduce a renormalized coupling constant

1

gr=

1

g+M

h21

ǫ, (9.163)

and rewrite the right-hand side as

h

gr− M

2hπlog πκδ. (9.164)

This has a pole at

κB =1

πδe2h

2π/Mgr , (9.165)

indicating a bound-state pole of energy EB = −h2κ2B/2M .We can now go to the limit of D = 2 dimensions and find that the pole term in

(9.161) has the form

ψB(xb)ψ∗B(xa)

ih

E −EB, (9.166)

with the normalized bound-state wave function

ψB(x) =κB√πK0(κB|x−X|). (9.167)

Notes and References

The wave functions derived in this chapter from the time evolution amplitude should be comparedwith those given in standard textbooks on quantum mechanics, such asL.D. Landau and E.M. Lifshitz, Quantum Mechanics , Pergamon, London, 1965. The chargedparticle in a magnetic field is treated in §111.The δ-function potential was studied via path integrals byC. Grosche, Phys. Rev. Letters, 71, 1 (1993).


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