Quantum Field Theory Victor Gurarie Fall 015 Lecture 9: Path Integrals in Quantum Mechanics
1 Path integral in quantum mechanics 1.1 Green s functions of the Schrödinger equation Suppose a wave function satisfies the Schrödinger equation: i t ψ = Ĥψ = 1 ) m q + V q) ψ. 1.1) Given ψt, q) and ψt 0, q 0 ), there is a function Gt, t 0 ; q, q 0 ) such that ψt, q) = dq 0 Gt, t 0 ; q, q 0 )ψt 0, q 0 ). 1.) Here Gt, t 0 ; q, q 0 ) is the Green s function of the Schrödinger equation. It satisfies This is required so that lim Gt, t 0 ; q, q 0 ) = δq q 0 ). 1.3) t t 0 lim ψt, q) = ψt 0, q). 1.4) t t 0 The Green s function also satisfies i t Gt, t 0 ; q, q 0 ) 1 ) m q + V q) G = 0. 1.5) so that ψt, x) satisfies the Schrödinger equation 1.1). We can argue that the Green s function depends on the time difference only Gt, t 0 ; q, q 0 ) = Gt t 0 ; q, q 0 ). 1.6) It follows that ψt, q ) = dq 1 Gt t 1 ; q, q 1 )ψt 1, q 1 ) = Take small δt. Then we can write dq 1 dq Gt t 1 ; q, q 1 )Gt 1 t 0 ; q 1, q 0 )ψt 0, q 0 ). 1.7) ψt 0 + nδt, q n ) = n 1 i=0 [Gδt; q i+1, q i )dq i ] ψt 0, q 0 ). 1.8) Think of q i as the position of the particle at the time t 0 +iδt. Then we discover an integral over paths the particle takes in its propagation. 1
1. Green s function over infinitesimal time interval For small δt, we can find G systematically, by employing 1.5) and writing time derivative as a difference for small time intervals Therefore, This gives i Gδt, q, q 1 ) δq q 1 ) δt = 1 ) + V q m q ) δq q 1 ). 1.9) Gδt; q, q 1 ) = 1 iδt 1 )) + V q m q ) δq q 1 ). 1.10) Gδt; q, q 1 ) = It follows from here that dp π eipq q 1) iδt Gnδt, q n, q 0 ) = )) dp p 1 iδt π m + V q 1) e ipq q 1 ) = 1.11) ) = 1 mπ q q 1 ) π iδt eim iδtv q δt 1 ). 1.1) p m +V q 1) [ n 1 i=1 [ )] n 1 n 1 e i q i+1 q i ) V q i=0 δt i )δt i=1 Gδt; q i+1, q i )dq i ] Gδt; q 1, q 0 ) = 1.13) ) m n. 1.14) πiδt We recognize that what sits in the exponential becomes, in the limit δt 0, the action of the particle n 1 i=0 qi+1 q i ) ) V q i )δt δt nδt 0 m q We write 1.13) this limin the following schematic way 1.3 Gaussian integrals GT ; q n, q 0 ) = qt )=qn V q) ) = S. 1.15) q0)=q 0 Dqt) e is. 1.16) Most calculations using path integrals are based on application of the following Gaussian integrals. The first one is this e 1 N m,n=1 Amnxmxn N i=1 dx i = π) N. 1.17) det A
Here A is a symmetric matrix if it were not symmetric, the sum in the exponential would symmetrize it as it automatically contains expressions such as x 1 x A 1 + A 1 ), in other words replacing A by a symmetric matrix Ãmn = A mn + A nm )/ does not change the value of the integral) and the second one is e 1 mn Amnxmxn+ N N m=1 Jmxm i=1 dx i = π) N e 1 N m,n=1 Jm[A 1 ] J n mn. 1.18) det A Both can be derived using the standard methods of manipulating integrals. We define Z[J] = e 1 mn Amnxmxn+ N m=1 Jmxm N i=1 dx i e 1 N = e 1 N m,n=1 Jm[A 1 ] J n mn. 1.19) m,n=1 Amnxmxn N i=1 dx i From the second integral we can calculate the following correlation functions x α x β = xα x β e 1 N m,n=1 Amnxmxn N i=1 dx i e 1 N m,n=1 Amnxmxn N i=1 dx i = Z[J] = [ A 1] J α J β αβ J=0 1.0) Moreover, the correlation functions of these type satisfy Wick s theorem x α x β x γ x δ = 4 Z[J] = [ A 1] [ ] A 1 J α J β J γ J δ +[ A 1] [ ] A 1 +[ A 1] [ ] A 1. αβ γδ αγ βδ αδ βγ J=0 1.1) For further applications is also important to rewrite these integrals using complex formalism. Suppose we have two variables, x and y which we combine into z = x + iy. Notation z = x iy is also often used. Integration over the complex plane z is defined by integrating over dx and dy. For example, d z e λ zz dxdy e λx +y ) = π λ. 1.) Sometimes the notation d z = dzd z is used. Sometimes a convention By extension e mn AN m,n=1 zmzn m That s the analog of 1.17). We can also show d z m = πn det A. 1.3) Z[J] = e mn AN m,n=1 zmzn+ m J mz m+j m z m) m d z m e mn AN m,n=1 zmzn m d z m = e J mn m[a 1 ] J n mn. 1.4) 3
The correlation function is then z α z β = zα z β e mn AN m,n=1 zmzn m d z m e mn AN m,n=1 zmzn m d z m = [ A 1] αβ. 1.5) The Wick theorem for complex variables states 4 Z[J] z α z β z γ z δ = J α J = [ A 1] [ ] A 1 β J γ J δ + [ A 1] [ ] A 1. 1.6) αγ βδ αδ βγ J=0 1.4 Quasiclassical expansion of the Green s function The path integral is especially well suited for studying quantum motion close to classical limit. An evolution operator and by extension, the Green s function) of a particle moving in a potential is given by xtf )=x f Gx f, x i, t f ) = i Dxt) e i tf x )=x i [ mẋ Ux) ]. 1.7) To compute this Green s function we use the stationary phase method. We take the action S S = tf [ ] mẋ Ux). 1.8) and minimize it with respect to all the trajectories xt) beginning at x i and ending in x f. This is equivalent to solving the equations of motion with these boundary conditions δs δxt) = x md U x = 0. 1.9) We call such trajectory x cl t), since is a classical trajectory connecting the points x i and x f. The zero order approximation to the Green s function is then Gx f, x i, t f ) e is cl. 1.30) A better approximation is achieved if one takes into account fluctuations around the classical trajectory. We take For small y, we write S[x cl + y] S cl + xt) = x cl t) + yt). 1.31) tf mẏ 1 4 U x y. 1.3) x=xcl
Introduce, to simplify notations, ωt) = 1 m U x. 1.33) x=xcl Then we can write S = S cl tf Then by analogy with 1.17) we find [ ] yt) m d + ωt) yt). 1.34) G e is 1 cl det [ m d ωt) ]. 1.35) We use the sign because there is also factor π) N involved - here N is the number of points when time is discretized. But these factors are pure numbers, do not depend on anything and can be ignored. Expressions like this one involving determinants of operators can be intimidating. But ultimately all they involve is solving the equation [ ] m d ωt) yt) = λ n yt) 1.36) with the boundary conditions y ) = yt f ) = 0 since xt f ) = x cl t f ) = x f and x ) = x cl ) = x i ). Then this determinans det [ ] m d ωt) = n λ n. 1.37) In some important cases is possible to compute this determinann the closed form. 1.5 Correlation functions For what follows we will also need to define time ordered correlation functions. These are 0 T ˆxt 1 )ˆxt )... ˆxt n ) 0. 1.38) There is a very natural way to define them using path integral. The expression for this correlation function reads 0 T ˆxt 1 )ˆxt )... ˆxt n ) 0 = Dxt) xt1 )... xt n ) e is Dxt) e is. 1.39) 5
Here the action S and the trajectory xt) is taken over a very large, ideally infinite, time interval, which includes all the times t 1, t,..., t n. The trajectory xt) is not fixed ats ends, unlike 1.7). This expression is one of the most common applications of functional integral, more common than 1.7). Its derivation is straightforward but long and tedious. We start by setting up a functional integral for a related but different correlation function. Assuming t 1 > t >... > t n, we will set up a path integral expression for x f e iĥt f t 1)ˆxe iĥt 1 t )ˆx... ˆxe iĥtn ) x i = xtf )=x f x )=x i Dxt) e is xt 1 )... xt n ). 1.40) The derivation of the right hand side proceeds in the same way as the derivation of 1.7). We are interested in computing the ground state expectation value. However, a very clever idea is to note thaf t f,, expanding x i = n c n n, only the ground state contributes. Same is done at t f. What this implies is that the left hand side of 1.40) can be rewritten as x f m m e iĥt f t 1)ˆxe iĥt 1 t )ˆx... ˆxe iĥtn ti) n n x i. 1.41) mn If t f and, then only the ground state contributes. Finally, we observe that 0 T ˆxt 1 )ˆxt )... ˆxt n ) 0 = Dxt) xt1 )... xt n ) e is Dxt) e is. 1.4) Here the path integral is taken over trajectories from to +, with ends which are not fixed anywhere because the ends are far away in infinite past or future). 1.6 Harmonic oscillator A very prominent application of this formalism is to a harmonic oscillator. its action is S = tf [ mẋ mω x ] [ tf = pẋ p m mω x ]. 1.43) For a harmonic oscillator the expansion 1.3) is actually exact, not approximate. And ωt) = ω. Everything said in the previous section applies, with the formalism becoming exact. 6
For future applications, let us rewrite the path integral using complex variables. Define a = 1 [ x mω + ip ], ā = 1 [ x mω mω ip ]. 1.44) mω One can check that the action can be rewritten as [ S = iā da ] ωāa. 1.45) It follows that 0 T ât f )â ) 0 = Dat) atf )ā ) e i [iā da ωāa] Dat) e i [iā da ωāa] 1.46) This is often called coherent state path integral. 7