The Path Integral Formulation of Quantum Mechanics

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1 The Path Integral Formulation of Quantum Mechanics Shekhar Suresh Chandra March 15, 2005 Prerequisite: Knowledge of the Lagrangian Formalism (which can be found on my site also). Thirty-one years ago, Dick Feynman told me about his sum over histories version of quantum mechanics. The electron does anything it likes, he said. It just goes in any direction at any speed,... however it likes, and then you add up the amplitudes and it gives you the wavefunction. I said to him, Youre crazy. But he wasnt. Freeman Dyson, 1980 [1] In the classical theory of mechanics, the motion of an object occurs only in is to undergo the least action, i.e. it abides the Hamilton s Principle of Least Action where the action is defined by δs δ t2 t 1 L(q(t), q(t))dt 0 (1) S t2 t 1 L(q(t), q(t))dt (2) Quantum Mechanically however, the quantum mechanical object explores every possible path in its motion. Richard Feynman developed this idea to formulate Quantum Mechanics in a different way (by using path integrals) and definition (2) played a critical role. This formulation is useful in the theory of Quantum Fields. Non-relativistic quantum mechanics is governed by the Schrodinger Equation i t Ψ(t)> Ĥ(ˆp, ˆq) Ψ(t)> Monash University. Shekhar.Chandra@spme.monash.edu.au 1

2 2 where Ψ(t)> is the state vector in Hilbert space. The formal solution 1 to this is Ψ(t)> e iĥt/ Ψ(0)> In the generalised co-ordinate system q, the above becomes q, t> e iĥt/ q > (3) Therefore the transitional amplitude for the particle at q propagating to q, i.e. the probability density of particle transiting from q to q is <q, t q, t> <q e iĥ(t t)/ q > (4) This is known as the Feynman Kernal or Propagator. Conducting the path integral formalism for the propagator, one can construct the Quantum Mechanical Transitional Amplitude from the classical Lagrangian of the system, without the need for Hilbert Space state vectors and noncommuting Operators. In order to do this, we must divide the time interval t t into small sub-intervals, i.e. t n t + n t and so t t N t where n 0, 1,..., N 1. We substitute this into the propagator to get <q e iĥn t/ q > (5) Now at each sub-interval, a set of basis states is introduced using the completeness relation 2 Therefore the propagator q n ><q n dq n I (6) <q e iĥn t/ q > <q e iĥ t/ Ie iĥ t/ Ie iĥ t/ I... q > Substituting for the identity I in above <q e iĥn t/ q >... dq N 1... dq <q e iĥ t/ q N 1 ><q N 1... q 1 ><q 1 e iĥ t/ q > Which can be re-written as <q e iĥn t/ q > N 1 n1 dq n <q e iĥ t/ q N 1 ><q N 1... q 1 ><q 1 e iĥ t/ q > (7) 1 proof can be found in appendix A 2 proof can be found in appendix B

3 3 The physical interpretation of the above equation is that the particle can take any path in its transition from (q, t) to (q, t ). Now the inner products, i.e. tranisition amplitudes (sometimes referred to as transfer matiices ) T (q n+1, q n ) <q n+1 e iĥ t/ q n > Can be simplified in some cases. We shall consider such a case of a free particle, i.e. when Ĥ ˆp2 2m (8) Therefore, the transition matrix becomes <q n+1 e iĥ t/ I q n > dp n 2π <q n+1 e iĥ t/ p n ><p n q n > (9) Here we have used a slightly different form of the completeness relation (one with normalisation constants kept for future convenience), namely of the form 1 2π p n ><p n dp n I (10) Using the property of state vectors, that F (Â) A> F (A) A> and relation (8) then Using this, the relation (9) becomes <q n+1 e iĥ t/ I q n > e iĥ t/ p n > e ip2 t/2m p n > dp n 2 2π e ip t/2m <q n+1 p n ><p n q n > (11) Each of the bra-kets in the above can be determined from the normalisation convention 3 Hence relation (11) becomes <q n+1 e iĥ t/ I q n > <q n p n > e ipnqn/ dp n 2 2π e ip t/2m e ipnqn+1/ e ipnqn/ (12) dp n 2 2π e ip t/2m e ipn(qn+1 qn)/ (13) The above relation is the form of a Gaussian. Making the change of variables 3 proof can be found in appendix C α t m (14) J 1 (q n+1 q n ) (15)

4 A FORMAL SOLUTION TO SCHRODINGER S EQUATION 4 Then the Gaussian Integral 4 is I Now the transfer matrix can be written Substituting back into the kernel <q n+1 e iĥ t/ q n > e 1 2 iαx2 +ijx dx 2πi α im 2π t ei t m 2 ij 2 e 2α ( qn+1 ) qn 2 t <q e iĥn t/ q > ( ) N/2 N 1 im 2π t ( im 2π t ) N/2 N 1 n0 n0 N 1 dq n n0 dq n e i t m 2 e i t m 2 N 1 n0 ( qn+1 qn ) 2 t (16) ( qn+1 qn t ) 2 (17) Now we proceed to take the limit as t 0 and hence N, the result being ( ) lim t 0 <q e iĥ(t t)/ i t 1 q > Dq(t)exp t 2 m q2 (t)dt Which is off the form of <q e iĥ(t t)/ q > ( ) i t Dq(t)exp L(q, q)dt t ( ) i Dq(t)exp S(q, q) (18) (19) Appendix A Formal Solution to Schrodinger s Equation Theorem 1 Show that q, t> e iĥt/ q > We start with the Schrodinger Equation i Ψ> Ĥ Ψ> t which can be writen as 1 Ψ> t Ψ> Ĥ i 4 worked through in appendix D

5 B PROOF OF THE COMPLETENESS RELATION 5 Notice that... Ψ> t d Ψ> Ψ 0> Ψ> iĥ dt (20) t 0 [ ] Ψ> log e ( Ψ>) iĥ [ ] t t (21) Ψ 0> t 0 log e ( Ψ>) log e ( Ψ 0 >) iĥ (t t 0) (22) ( ) Ψ> log e iĥ Ψ 0 > (t t 0) (23) Ψ> Ψ 0 > e iĥ (t t0) (24) Ψ> e iĥ (t t0) Ψ 0 > (25) B Proof of the Completeness Relation Theorem 2 Show that q ><q dq I Take any arbitary ket, say ξ >, then check for consistency ξ > I ξ > (26) Here we have used the sifting property of the delta function. dq q ><q ξ > (27) dq q > δ(q ξ) (28) ξ > (29) C Proof of Normalisation Convention Theorem 3 Show that <q n p n > <p n q n > e ipnqn/ We begin by writing the state vector as Ψ> I Ψ>

6 C PROOF OF NORMALISATION CONVENTION 6 Then we substitute the identity with the co-ordinate completeness relation Ψ> dq n q n ><q n Ψ> Now we project the state vector onto the momentum representation p and we get <p n Ψ> dq n <p n q n ><q n Ψ> (30) Since < q n Ψ > is the co-ordinate representation of the state vector, we denote it by Ψ q >. Similarly < p n Ψ > is the momentum representation of the state vector, we denote it by Ψ p >. Hence equation 30 becomes Ψ p > dq n <p n q n > Ψ q > (31) The above is of the form of a Fourier Transform, as the co-ordinate space is being transformed into momentum or reciprocal space. Fourier Transform is of the form F [f(q)] Therefore, in order for equation 31 to be true f(q)e ipq/ dq <p n q n > e ipnqn/ (32) For the case of < q n p n >, it will be the coomplex conjugate of equation 32. This can be shown by taking the momentum completeness relation instead of the co-ordinate completeness relation, giving Ψ> dp n p n ><p n Ψ> Projecting the state vector onto co-ordinate representation to acquire the momentum equivalent relation to equation 30, namely <q n Ψ> dp n <q n p n ><p n Ψ> Re-writing the above in a similar way for the <p n q n > case, giving Ψ q > Now this is the inverse Fourier Transform, which has the form F 1 [f(p)] Hence dq n <q n p n > Ψ p > (33) f(p)e ipq/ dp <q n p n > <p n q n > e ipnqn/ (34)

7 D THE GAUSSIAN INTEGRAL 7 D The Gaussian Integral Theorem 4 Show that I e 1 2 iαx2 +ijx dx 2πi α ij 2 e 2α References [1] Dyson F. Some Strangeness in the Proportion, edited by H. Woolf AddisonWesley, Reading, MA, 1980, p. 376.