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Path Integrals in Quantum Mechanics Path Integrals in Quantum Mechanics Joachim Ankerhold and Bj¨ornKubala WS 2014/15 Be aware: May still contain typos December 11, 2014 I. Introduction II. Schr¨odingerquantum mechanics 1. Time evolution and the propagator 2. The free particle III. Path integral representation 1. Real-time propagator 2. Canonical operator and partition function 3. Ehrenfest's theorem and functional derivatives 4. Correlation functions IV. Gaussian path integrals and harmonic oscillator 1. Harmonic oscillator and normalization 2. Gaussian path integrals and normalization V. Semiclassics 1. Stationary phase/Saddle point Treatment 2. Examples 3. Tunneling 2 I. Introduction The conventional formulation of quantum mechanics as it is taught to students is based on the Schr¨odingerequation. This is in some contrast to the actual situation in theo- retical physics, where all modern developments make extensive use of the path integral formalism, in particular in modern field theories in atomic, condensed matter, statistical, and polymer physics, in cosmology, elementary particle physics and even in econophysics. The first who introduced 'a sum over paths' was N. Wiener who worked with Wiener measures to analyze problems in statistical physics and Brownian motion. This idea was extended by Paul Dirac in 1933/34 ( Physical Journal of the Soviet Union 3, 64 (1933) to quantum mechanics. However, it was R. Feynman who developed the full method in the 1940s and described it in detail in his seminal paper 'Space-Time Approach to Non- Relativistic Quantum Mechanics'. Rev. Mod. Phys. 20, 367 (1948). Since then the path integral representation of quantum mechanics has been put forward, generalized, and laid on a firm mathematically basis. It was clear from the very beginning that path integrals provide an alternative way of understanding quantum mechanics which allows for calculations, approximations, and formal manipulations that are very difficult, if not impossible, to achieve within the Schr¨odingerrepresentation. In a way, the two framework are complementary: to obtain wave functions and eigenergies, the Schr¨odingerequation is the most convenient tool, for semiclassical approximations and many-body problems path integrals are the right tools. In this lecture we will learn the basic ingredients of the formalism starting with the formulation as it was originally introduced by Feynman. This is not a math course in functional integration meaning that we have to skip all mathematical subtleties and instead follow a more 'pragmatic' approach. However, we will refer to the literature for those who are interested. There are many books which address the path integral formulation. Here is a selection: • Feynman, R. P. and Hibbs, A. R. (1965). Quantum Mechanics and Path Integrals. New York: McGraw-Hill. ISBN 0-07-020650-3. • Schulman, Larry S. (1981). Techniques & Applications of Path Integration. New York: John Wiley & Sons. ISBN 0486445283. 3 • Glimm, James, and Jaffe, Arthur (1981). Quantum Physics: A Functional Integral Point of View. New York: Springer-Verlag. ISBN 0-387-90562-6. • Hagen Kleinert (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (4th ed.). Singapore: World Scientific. ISBN 981-238-107-4. • About mathematical foundations: DeWitt-Morette, Ccile (1972). 'Feynman's path integral: Definition without limiting procedure'. Communication in Mathematical Physics 28 (1): 47; Albeverio, S. & Hoegh-Krohn. R. (1976). Mathematical Theory of Feynman Path Integral. Lecture Notes in Mathematics 523. Springer- Verlag. ISBN 0-387-07785-5. 4 II. Schr¨odingerquantum mechanics II.1. Time evolution and the propagator The dynamics of a quantum system with Hamiltonian H follows from the Schrdinger¨ equation d Hj (t)i = i j (t)i : (II.1) ~dt i − Ent For time-independent Hamiltonians one finds from j (t)i = e ~ jφni that the eigen- states are the stationary solutions to Hjφni = Enjφni (II.2) and that they form a complete basis set of the Hilbert space with hφnjφmi = δn;m.A classical phase space with conjugate variables q and p with Poisson brackets fq; pg = 1, is quantized by imposing [q; p] = i~. A formal solution to the time-dependent Schr¨odingerequation is given for t ≥ 0 by j (t)i = G(t) j (0)i ;G(t) = e−iHt=~ ; (II.3) where the propagator has the spectral representation X −iEnt=~ G(t) = e jφnihφnj ; t ≥ 0 (II.4) n and thus carries the same information as the solutions of the time-independent Schr¨odinger equation. Its formal Fourier transform 1 X jφnihφnj G~(E) = (II.5) i E − E − io+ n n gives direct access to the energy spectrum (here o+ enures that the contour integration back to time-space has only contributions for t ≥ 0). The kernel can thus also be considered as the central object in quantum mechanics 5 (instead of the wave functions). It has the following properties 1. It obeys a semi-group identity, i.e., G(t; 0) = G(t; t0) G(t0; 0) with 0 ≤ t0 ≤ t (II.6) 2. In position representation, this identity reads G(qf ; qi; t) ≡ hqf jG(t)jqii Z 0 0 = dqG(qf ; q; t − t ) G(q; qi; t ) (II.7) Remark: Note that the propagator depends on time differences only for time- independent Hamiltonians. 3. It easily follows that d i G(t; t0) = HG(t; t0) with G(t0; t0) = 1 (II.8) ~dt 0 0 0 4. The retarded propagator Gr(t; t ) = θ(t − t ) G(t; t ) obeys in position space i~@tf − H(qf ; −i~@qf ) G(qf ; qi; tf ; ti) = i~δ(tf − ti) δ(qf − qi) : (II.9) and is thus the Greens-function of the Schr¨odingerequation. Due to these properties, it was Feynman's idea to formulate quantum mechanics only based on a representation of the propagator. This then leads to the 'integral' form of the Schr¨odingerequation in terms of path integrals. II.2. The free particle As an example we consider the free particle with H = p2=2m for which the propagator can easily be calculated analytically. From 2 − i p^ G(qf ; qi; t) = hqf je ~ 2m jqii (II.10) we obtain by inserting a complete set of momentum eigenstates 1 = R dp jpihpj with 1 i pq hqjpi = p e ~ (II.11) 2π~ 6 the following integrals Z 1 2 − i p G(qf ; qi; t) = dp hqf jpi e ~ 2m hpjqii −∞ 1 1 Z i p2 i − p(qf −qi) = dp e ~ 2m e ~ 2π~ −∞ r m i m 2 (qf −qi) = e ~ 2 t : (II.12) 2πi~t Remark: The exponent coincides with the classical action of a free particle running in time t from qi to qf , i.e., Z t 2 q_(s) m 2 S(qf ; qi; t) = ds = (qf − qi) : (II.13) 0 2m 2 t The corresponding path is s q (s) = q + (q − q ) (II.14) cl i f i t m t and the action is conveniently obtained from S(qf ; qi; t) = 2 q(s)_q(s) 0. 7 III. Path integral `ala Feynman III.1. Real-time propagator We consider a system with Hamiltonian p2 H = + V (q) 2m = T + V (III.1) with conjugate operators [p; q] = −i~. Then, the whole path integral business starts with N G(t) = e−iH=~ ; = t=N 2 N = e−iT=~ e−iV=~ e−C=~ ; (III.2) where C is a 'correction' operator according to the Baker-Campbell-Hausdorff formula. The correction operator is a series in with its lowest order term being C = 2[T;V ] + O(3). We thus think of dividing the time interval [0; t] into N − 1 equidistant segments of length , where is assumed to be much smaller than any other relevant time scale. In the very end, we will consider limit, where ! 0 but N = t = const:. Mathematically, this 'factorization' is a special case of the Trotter formula N N eA+B ≡ eA=N+B=N = lim eA=N eB=N (III.3) N!1 which holds for bounded operators but can also be proven for unbounded operators under certain conditions. We do not discuss this subtle issue any further but proceed with its application for the propagator in position representation. 8 Accordingly, G(qf ; qi; t) ≡ hqf jG(t)jqii N −iH=~ = hqf j e jqii −iH=~ −iH=~ = hqf j e ::: e jqii | {z } N-times Z N−1 ! N Y Y −iH=~ = dqk hqkje jqk−1i ; (III.4) k=1 k=1 R where we inserted N − 1 resolutions of the identity dqk jqkihqkj = 1; k = 1;:::;N − 1 and put q0 = qi and qf = qN , see Fig. III.1. One may then write 2 −iT=~ −iV=~ −C=~ hqkjG()jqk−1i = hqkje e e jqk−1i −iT=~ −iV (qk−1)=~ ≈ hqkje jqk−1i e ; (III.5) where the second line becomes a strict identity in the limit N ! 1 with N = t = const. q q1 q2 q3 qN 1 ¡ qf qi ² 2² 5² t = N ² Time Figure III.1.: Trotter discretization for the real-time propagator according to (III.4). What remains to do is, to plug in the expression for the propagator of a free particle in position space that we derived already in (II.12). For the above matrix element it reads r σ 2 hq je−iT=~jq i = eiσ(qk−qk−1) (III.6) k k−1 iπ 9 with σ = m=2~. Accordingly, Z N−1 r ! ( " N 2 #) Y σ i X m(qk − qk−1) G(q ; q ; t) = dq exp − V (q ) : (III.7) f i k iπ 2 k−1 k=1 ~ k=1 The somewhat subtle part is now to perform the limit N ! 1, N = t. In the exponent we put (qk − qk−1)/ ! q_(s) and assume that in this limit the fqkg give rise to paths q(s); 0 ≤ s ≤ t with boundary conditions q(0) = q0 ≡ qi and q(t) = qN ≡ qf .
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