Brizard A. An Introduction to Lagrangian Mechanics

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Appendix A

Notes on Feynman’s Quantum

Mechanics

A.1 Feynman postulates and quantum wave function

Feynman

makes use of the Principle of Least Action to derive the Schroedinger equation

by, ﬁrst, introducing the following postulates.

Postulate I: Consider the initial and ﬁnal states a and b of a quantum system and

the set M of all paths connecting the two states. The conditional probability amplitude

K(b|a) of ﬁnding the system in the ﬁnal state b if the system was initially in state a is

expressed as

K(b|a) ≡

φ[X], (A.1)

where the summation is over all paths X(t) from a to b and the partial conditional proba-

bility amplitude associated with path X(t)is

φ[X] ∝ exp



¯h

S[X]



, (A.2)

with S[X] corresponding to the classical action for this path. If we take the two states

to b e inﬁnitesimally close to each other, i.e., whenever the points a ≡ x

(at time t

) and

b ≡ x

+∆x (at time t

+∆t) are inﬁnitesimally close, Eqs. (A.1)-(A.2) yield

K(x

+∆x, t

+∆t|x

exp

¯h

m(∆x)

2∆t

− ∆tU



∆x

∆t



, (A.3)

The material presented in this Appendix is adapted from the book Quantum Mechanics and Path

Integrals by R.P. Feynman and A.R. Hibbs (McGraw-Hill, New York, 1965).

165

166 APPENDIX A. NOTES ON FEYNMAN’S QUANTUM MECHANICS

where A is a normalization constant and the classical action integral

S[x]=

+∆t





− U(x, t)





dt =

m(∆x)

2∆t

− U



∆x

∆t



∆t

is a function of ∆x and ∆t as well as the initial space-time point (x

); here, the potential

energy U is evaluated at the mid-point between x

and x

at a time t

+∆t/2.

Postulate I provides the appropriate explanation for the mystery behind the Principle

of Least Action (“act locally, think globally”). Indeed, in the classical limit (¯h → 0), we

ﬁnd that the variations δX around the paths X(t) which are far away from the physical

path x(t) yield changes δS for which δS/¯h is large. Consequently, such contributions tend

to average out to zero because of the corresponding wild oscillations in φ[X]. On the other

hand, for variations δX near the physical path x(t), we get δS ∼ 0 (to ﬁrst order) and,

consequently, paths X(t) for which S[X] is within ¯h of S

will contribute strongly. The

resulting eﬀect is that only paths in the neighborhood of the physical path x(t) have a

nonvanishing probability amplitude. In the strict limit ¯h → 0, the only such path with a

nonvanishing probability amplitude is the physical path.

Postulate II: The quantum wave function ψ(x, t) is deﬁned as the probability ampli-

tude for the particle to be at the location x at time t, i.e., ψ(x, t) ≡ K(x, t|•), where we

are not interested in the previous history of the particle (its previous location is denoted

by •) but only on its future time evolution. The Second Postulate states that the integral

equation relating the wave function ψ(x

) to the wave function ψ(x

) is given as

ψ(x

) ≡

∞

−∞

K(x

) ψ(x

). (A.4)

If we set in Eq. (A.4): t

= t and t

= t + , x

= x and x

= x + η, then for small enough

values of , the conditional probability amplitude (A.3) can be used in (A.4) to yield

ψ(x, t + ) ≡

∞

−∞

dη K(x, t + |x + η,t) ψ(x + η,t) (A.5)

∞

−∞

dη

exp

¯h

mη

2

− U(x + η/2,t+ /2)

ψ(x + η, t),

where a time-dependent potential U(x, t) is considered.

A.2 Derivation of the Schroedinger equation

We will now derive the Schroedinger equation by expanding both sides of Eq. (A.5), up to

ﬁrst order in  (and neglect all higher powers). First, on the left side of Eq. (A.5), we have

ψ(x, t + )=ψ(x, t)+

∂ψ(x, t)

∂t

. (A.6)

A.2. DERIVATION OF THE SCHROEDINGER EQUATION 167

Next, on the right side of Eq. (A.5), we note that the exponential

exp



2¯h





oscillates wildly as  → 0 for all values of η except those for which mη

/(2¯h) ∼ 1. We,

therefore, conclude that the contribution from the integral in Eq. (A.5) will come from

values η = O(

1/2

) and, consequently, we may expand the remaining functions appearing

on the right side of Eq. (A.5) up to η

. Thus, we may write

ψ(x + η, t)=ψ(x, t)+η

∂ψ(x, t)

∂x

∂

ψ(x, t)

∂x

while

exp



−

i

¯h

U(x + η/2,t+ /2)



=1−

i

¯h

U(x, t).

Expanding the right side of Eq. (A.5) to ﬁrst order in , therefore, yields



1 −

i

¯h

U(x, t)



ψ(x, t)+





∂ψ(x, t)

∂x



2 I



∂

ψ(x, t)

∂x

, (A.7)

where A = I

and

≡

∞

−∞

dη η

−aη

,a≡

2i¯h

with I

π/a, I

= 0, and I

=1/2a =(i¯h/m) .

Lastly, we ﬁnd that the terms of ﬁrst order in  in Eqs. (A.6) and (A.7) must be equal,

and we obtain

∂ψ(x, t)

∂t

= −

¯h

U(x, t) ψ(x, t)+

i¯h

∂

ψ(x, t)

∂x

i¯h

∂ψ(x, t)

∂t

= −

¯h

∂

ψ(x, t)

∂x

+ U(x, t) ψ(x, t). (A.8)

This equation is known as the Schroedinger equation and it describes the time evolution

of the wave function ψ(x, t).