Path-Integral Quantum MechanicsCooper JohnstonLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYvI. Abstract Many mathematical formulations of Quantum Mechanics have been developed with the goal of describing particular phenomena in a convenient way. The wave formulation is useful for capturing the phase behavior of quantum systems, characteristic of the wave-particle duality and matrix mechanics lends itself to describing quantum systems using matrix algebra to capture the non-commutative nature of measurements. While these two formulations are the most commonly taught, due in large part to their mathematical approachability, there exist other formulations more clearly describe other phenomena. The Path-Integral formulation of Quantum Mechanics succinctly captures the time evolution of quantum systems by generalizing the Principle of Least Action, arguably the most fundamental principle undergirding Classical Mechanics. Path-Integral Quantum Mechanics is centered on an object called the Propagator, K(x'',t'';x',t'), which is a matrix element of the time evolution operator from t' to t'' from an initial state x' to a final state x''. With clever use of matrix calculus, we will find that the Propagator is closely related to the Classical Lagrangian and acting it against some initial quantum state results in an integral over all paths weighted by the complex exponential of the Lagrangian.II. Introduction Many introductions to the Path-Integral formulation begin with the same story and far be it for me to flaunt that tradition. The story begins with Richard Feynman sitting bored in a Physics class at MIT. The lecturer then turned his attention to the strangely quiescent nature of particles as both wave and particle, piquing the interest of Feynman. "The double slit experiment demonstrates that particles, thought to be discrete and indivisible, display some wave-like character, interfering with themselves as they propagate through the slits. When the system is otherwise not measured, this effect forces one to conclude that the particle passes equally through both." Upon hearing this notion, Feynman asks what happens if a third slit is added. "Well certainly we will have that the particle passes in some measure through all of the paths; in the case of very close packed slits, it passes equally through all." Feynman then asks about adding another slit and is met with a similar, albeit more annoyed, response. After thinking for a moment, Feynman asks a final question which is met with no reply at all: "What if one were to have infinite slits side by side, packed together and then stacked infinitely many of these sheets between the source and the detector?" This is often cited as the moment that the path integral was first born.III. TheoryIII.I. Deriving the Path Integral from the Time-Dependent Schr\303\266dinger Equation We begin the derivation of the Path Integral of the eponymous formulation of Quantum Mechanics from the familiar Time-Dependent Schr\303\266dinger Equation (TDSE) with a general Hamiltonian for a particle defined as the sum of the kinetic energy operator and an arbitrary potential. Note these expressions can be made general for higher dimensional systems with the Einstein Summation Convention.(i\342\204\217\342\210\202t - \342\204\213)\317\210=0LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2MFEifkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLRi9GMkY1Rjg=\342\204\213=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\342\210\202x2+V(x)We then choose an arbitrary time dependent quantum state and project it onto the position basis to construct and arbitrary wavefunction for this system. The dynamics of the system are governed by the TDSE and so we extract the time dependence of the quantum state using the Time Evolution operator, leaving the quantum state to be evaluated at t=0.\317\210(x',t)=\342\214\251x'|\317\210(t)\342\214\252=\342\214\251x'|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|\317\210(t=0)\342\214\252Using the completness relation of the position basis, we may expand the initial wavefunction in position eigenstates.\317\210(x',t)=\342\214\251x'|\317\210(t)\342\214\252=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\342\214\251x'|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|x\342\214\252\342\214\251x|\317\210(t=0)\342\214\252We now define the Propagator to be the expectation value of the time evolution operator seen in the expression above.K(x',t;x,0)=\342\214\251x'|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|x\342\214\252Thus, we may rewrite the time dependent state using the Propagator.\317\210(x',t)=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K(x',T;x,0)\342\214\251x|\317\210(t=0)\342\214\252
We will now shift our focus to the Propagator to develop its relationship with the Classical Lagrangian. We first consider evolving a state from an initial time t=0 to a final time t=T. we will subdivide this interval into N subintervals of uniform width. This yields a product of N Time Evolution operators, each over a time interval of t/N.K(x',T;x,0)=\342\214\251x'|LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LlEhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZFLUYjNiYtSSVtc3VwR0YkNiUtSShtZmVuY2VkR0YkNiYtRiM2Ji1GSzYlLUYsNi9RLyZFeHBvbmVudGlhbEU7RicvJSVzaXplR1EjMTRGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZDL0ZHUSwwLjExMTExMTFlbUYnLUYjNictRiw2L1EqJnVtaW51czA7RidGV0YvRjJGNUY3RjlGO0Y9Rj9GQS9GRFEsMC4yMjIyMjIyZW1GJy9GR0Zcby1JJm1mcmFjR0YkNigtRiM2Ki1JI21pR0YkNidRImlGJ0ZXLyUnaXRhbGljR1EldHJ1ZUYnRi8vRjNRJ2l0YWxpY0YnLUYsNi5RMSZJbnZpc2libGVUaW1lcztGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZDRkYtRmRvNidRJyZIc2NyO0YnRldGZ29GL0Zqb0ZccC1GZG82J1EiVEYnRldGZ29GL0Zqb0ZXRi9GMi1GIzYoRistRiM2KC1GZG82J1EnJmhiYXI7RidGVy9GaG9GMUYvRjJGXHAtRmRvNidRIk5GJ0ZXRmdvRi9Gam9GV0YvRjJGK0ZXRi9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZXEvJSliZXZlbGxlZEdGMUZXRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGV0YvRjJGV0YvRjJGXXFGanFGV0YvRjJGK0ZXRi9GMg==|x\342\214\252 We will now insert a complete position basis between each factor of the Time Evolution operator, resulting in N-1 copies of the position basis being inserted.K(x',T;x,0)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbW9HRiQ2LlErJkludGVncmFsO0YnLyUlc2l6ZUdRIzE0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctRiw2LlEoJmN0ZG90O0YnRi9GMkY1RjgvRjtGN0Y9L0ZARjdGQUZDRkVGSEYrLUYsNi5RIn5GJ0YvRjJGNUY4Rk1GPUZORkFGQ0ZFRkgtRiw2LlEwJkRpZmZlcmVudGlhbEQ7RidGL0YyL0Y2USZ1bnNldEYnL0Y5RlYvRjtGVi9GPkZWL0ZARlYvRkJGVi9GREZWRkVGSC1JI21pR0YkNiZRInhGJ0YvLyUnaXRhbGljR0Y8L0YzUSdpdGFsaWNGJy1GaG42J1EiTkYnRi9GW28vJSpzdWJzY3JpcHRHRjxGXW8tRiw2L1EoJm1pbnVzO0YnRi9GYm9GMkY1RjhGTUY9Rk5GQUZDL0ZGUSwwLjIyMjIyMjJlbUYnL0ZJRmhvLUkjbW5HRiQ2JlEiMUYnRi9GYm9GMkZKLUZobjYmUSNkeEYnRi9GW29GXW8tRmhuNidGXXBGL0Zbb0Zib0Zdb0YvRjI=\342\214\251x'|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|xN-1\342\214\252 \342\214\251xN-1|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|xN-2\342\214\252 \342\213\257 \342\214\251x1|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|x\342\214\252Similarly, we will insert N momentum bases, each one prior to a time evolution operator. in addition, we will take the infinite limit of this expression as N tends to infinity. K(x',T;x,0)=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\342\214\251x'|pN\342\214\252 \342\214\251pN|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|xN-1\342\214\252 \342\214\251xN-1|pN-1\342\214\252 \303\227\342\214\251pN-1|LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2L1EvJkV4cG9uZW50aWFsRTtGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xMTExMTExZW1GJy1JJW1yb3dHRiQ2Mi1GLDYvUSomdW1pbnVzMDtGJ0YvRjJGNS9GOVEmZmFsc2VGJy9GPEZUL0Y+RlQvRkBGVC9GQkZUL0ZERlQvRkZGVC9GSFEsMC4yMjIyMjIyZW1GJy9GS0Zmbi1JJm1mcmFjR0YkNigtRk42JS1JI21pR0YkNiZRImlGJ0YvRjJGNUYvL0Y2USdub3JtYWxGJy1GTjYlLUZebzYmUSkmaHNsYXNoO0YnRi9GMkY1Ri9GYW8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl1wLyUpYmV2ZWxsZWRHRlQtRl5vNiZRLyZIaWxiZXJ0U3BhY2U7RidGL0YyRjUtRiw2L1EifkYnRi9GMkY1RlNGVUZWRldGWEZZRlpGRy9GS0ZJLUZONiotRiw2LlEhRicvJStleGVjdXRhYmxlR0ZURmFvRlNGVUZWRldGWEZZRlpGR0ZocEZbcUZbcUZbcUZbcS1GaW42KC1GTjYmLUZebzYmUSJURidGL0YyRjVGL0YyRjUtRk42JS1GXm82J1EiTkYnRi9GMkZecUY1Ri9GYW9GaG9GW3BGXnBGYHBGL0Zhb0ZbcUZbcUZbcUZbcUZbcUZbcUZbcUZbcUZbcUYvRmFvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw==|xN-2\342\214\252 \342\213\257 \342\214\251x1|p1\342\214\252 \342\214\251p1|LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2L1EvJkV4cG9uZW50aWFsRTtGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xMTExMTExZW1GJy1JJW1yb3dHRiQ2MC1GLDYvUSomdW1pbnVzMDtGJ0YvRjJGNS9GOVEmZmFsc2VGJy9GPEZUL0Y+RlQvRkBGVC9GQkZUL0ZERlQvRkZGVC9GSFEsMC4yMjIyMjIyZW1GJy9GS0Zmbi1JJm1mcmFjR0YkNigtRk42JS1JI21pR0YkNiZRImlGJ0YvRjJGNUYvL0Y2USdub3JtYWxGJy1GTjYlLUZebzYmUSkmaHNsYXNoO0YnRi9GMkY1Ri9GYW8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl1wLyUpYmV2ZWxsZWRHRlQtRl5vNiZRLyZIaWxiZXJ0U3BhY2U7RidGL0YyRjUtRiw2L1EifkYnRi9GMkY1RlNGVUZWRldGWEZZRlpGRy9GS0ZJLUZONiktRiw2LlEhRicvJStleGVjdXRhYmxlR0ZURmFvRlNGVUZWRldGWEZZRlpGR0ZocEZbcUZbcUZbcS1GaW42KC1GTjYmLUZebzYmUSJURidGL0YyRjVGL0YyRjUtRk42JS1GXm82J1EiTkYnRi9GMkZecUY1Ri9GYW9GaG9GW3BGXnBGYHBGL0Zhb0ZbcUZbcUZbcUZbcUZbcUZbcUZbcUYvRmFvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw==|x\342\214\252As the expression currently stands, our previous steps have resulted in significant compliaction. This has been for good reason. The Hamiltonian is a sum of a kinetic energy operator, which commutes with the momentum operator, and a position dependent potential, which commutes with the position operator. Notably, these terms do not commute with each other so we cannot separate them into two exponential factors. This problem is solved by taking N to infinity, guaranteed by the Trotter Product Theorem which states that for two noncommuting operators A and B: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We know focus only on factors which are subject to the integral with measure dp1 in the previous expression for the Propagator for clarity. We now expand the Hamiltonian into a product of two exponentiated operators and evaluate them. We also note that the inner product of a position state and a momentum state may be evaluated by noting that the one transforms into the other via a Discrete Fourier Transform with appropriate normalization constant. The variable x' is introduced to denote that V(x') is position dependent operator and V(x) is the eigenvalue corresponding to 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The integral to be performed is over all of momentum space, so it resembles a Gaussian integral, albeit with a shifted argument to provide the term linear in momentum. The complication, however, is that the exponential is complex. There are several ways in which the situation may be salvaged. Firstly, this function is very oscillatory as the absolute value of momentum grows, as demonstrated explicitly for the real part of the integrand in the figure below. Note that many of the constants are set to one for clarity and to show the behavior of this functional form.with(plots):
Integrand := exp(-i(x^2+x)):RealPart := cos(x^2+x):ImaginaryPart := -sin(x^2+x):a := plot(RealPart,x=-3*Pi..3*Pi,color=red,title="Real Part of Integrand",size=[400,400]):
b := plot(ImaginaryPart,x=-3*Pi..3*Pi,color=blue,title="Imaginary Part of Integrand",size=[400,400]):display(a);display(b);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYvAs seen in the plots, the functions are highly oscillatory as we move away from zero momentum. Heuristically, we can argue that the oscillations at large values will contribute equally and oppositely to the area under the curve, strongly indicating that there is a finite result for this integral. The most straightforward method for evaluating this integral is agnostic to the domain of its arguments except for the imaginary prefactor which is removed by factoring the exponential. Completing the square then yields a tractable integral which nothing more than a shifted Gaussian.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=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We now shift our focus back to the expression for the entire propagator and reconstruct the the propagator using the previous expression.K(x',T;x,0)=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Note that for brevity of notation, the initial and final states are understood to be the smallest and largest indexed x variables, respectively. We will now do two things: firstly we will evaluate the limit as N goes to infinity of the exponentials and second we will define a functional integral which compresses the notation to clarify what this fomulation states. By taking the large N limit, the difference quotient in the exponential becomes a time derivative of position. Additionally, we also have a factor of T/N in the exponential which is becoming increasingly small. The product of these exponentials then sums the contributions of these small subintervals of time, essentially forming a time integral in the exponential. This finally yields the Propagator which functionally integrates over every possible path a particle can take from the initial state to the final state over the time interval [0,t].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K(x',T;x,0)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2L1ErJkludGVncmFsO0YnLyUlc2l6ZUdRIzE0RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTS1GIzYnLUkjbWlHRiQ2J1EieEYnRjIvJSdpdGFsaWNHRkJGNS9GOVEnaXRhbGljRidGMkY1LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjgtRiM2KEZSLUYvNi9RIidGJ0YyRjVGOC9GPEY3L0Y/RjcvRkFGNy9GREY3L0ZGRjcvRkhGNy9GSkY3L0ZMUSwwLjExMTExMTFlbUYnRk5GMkY1RlpGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGZ28tRlM2J1EnJkRzY3I7RidGMkZWRjVGWC1JKG1mZW5jZWRHRiQ2KC1GIzYoRlItRl5wNiYtRiM2Jy1GUzYnUSJ0RidGMkZWRjVGWEYyRjVGWkY4RjJGNUY4RjJGNUZaRjhGMkY1RjgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRi82L1EifkYnRjJGNUY4RlxvRl1vRl5vRl9vRmBvRmFvRmJvRktGTi1JJW1zdXBHRiQ2JS1GUzYnUSJlRidGMkZWRjVGWC1GIzYsLUkmbWZyYWNHRiQ2KC1GUzYnUSJpRidGMkZWRjVGWC1GIzYnLUZTNidRKCYjODQ2MztGJ0YyL0ZXRjdGNUY4RjJGVkY1RlgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRltzLyUpYmV2ZWxsZWRHRjctRiw2Jy1GIzYnLUYvNi5RIUYnRjVGOEZcb0Zdb0Zeb0Zfb0Zgb0Zhb0Zib0ZLRk5GLkYyRjVGOC1GIzYmLUkjbW5HRiQ2JkZnb0YyRjVGOEYyRjVGOC1GIzYmLUZTNidRIlRGJ0YyRlZGNUZYRjJGNUY4RmVvRmhvRl9xLUYvNi9RMCZEaWZmZXJlbnRpYWxEO0YnRjJGNUY4RjtGPi9GQUY9RkMvRkZGPUZHRklGS0ZORmZwLUZecDYmLUYjNistRltyNigtRlM2J1EibUYnRjJGVkY1RlgtRiM2Jy1GanM2JlEiMkYnRjJGNUY4RjJGVkY1RlhGZnJGaXJGXHNGXnMtRmNxNiUtSSZtb3ZlckdGJDYlRlItRi82MFEiLkYnRjJGNS8lLHBsYWNlaG9sZGVyR0ZCRjhGO0Y+RmR0RkNGZXRGR0ZJRktGTi9GSkZCRl91RmVvLUYvNi9RKCZtaW51cztGJ0YyRjVGOEZcb0Zdb0Zeb0Zfb0Zgb0Zhb0Ziby9GTFEsMC4yMjIyMjIyZW1GJy9GT0Zjdi1GUzYnUSJWRidGMkZWRjVGWC1GXnA2Ji1GIzYnRlJGMkZWRjVGWEYyRjVGOEYyRlZGNUZYRjJGNUY4RjJGVkY1RlhGZW9GZHNGMkY1RlpGOA==This final formulation of the propagator is the Feynman Path Integral for which this formulation is named. The physically inclined then observe that the integrand in the exponential function in exactly the classical action of the over a particular path subject to a potential V(x). Taking this into account, we can then write the most common expression for the Feynman Path Integral.K(x',T;x,0)=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The importance of this final result cannot be overstated. This derivation does not appeal to classical mechanics and yet we see that the classical action for a particular dynamical path through quantum phase space introduces a phase to the corresponding path in the sum over all paths. Thus, in a very real sense, we see that a system evolves according to all possible paths from the initial state to the final state where the contribution of each path is equal but with varying complex phase corresponding to the Classical action of that path. This, at first, feels very much in opposition to the Classical Principle of Least Action where a system evolves by a single unique path which is an extremal function of the Action. However, what we find is that, as system size grows, the possible paths have actions which tend to be large compared to \342\204\217. III.II. Properties of the PropagatorWe begin once again with the familiar notation of quantum mechanics of an inner product of two time dependent state |x',t'\342\214\252 and |x'',t''\342\214\252 where we assume that t'<t''. Often in Quantum Mechanics, we want to calculate the probability of a system in a particular state being found in either the same state or another state at some future time subject to the dynamics imposed by the TDSE. In many cases, we use the time evolution operator to accomplish this, but the operator is in a sense a black box. It doesn't explicitly lay out the dynamics of the system and describe how its evolution comes to pass as a result of physical principles, rather it takes in a state and spits out another state. The Feynman Path Integral makes the underlying source of the dynamics explicit by invoking a generalized Least Action Principle. We begin by rewriting the inner product of these states using the time evolution operator. This allows the transition amplitude to be rewritten as the Propagator.\342\214\251x'',t''|x',t'\342\214\252=\342\214\251x''|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|x'\342\214\252=\342\214\251x''|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|x'\342\214\252=K(x'',t'';x',t')Consider evaluating the Propagator where the initial and final state are the same. We define the initial time to be 0 and let t be the time at which we calculate the probability amplitude.K(x',t;x',0)=\342\214\251x'|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|x'\342\214\252We suppose that the system has discrete energy quantization and insert the complete basis of energy eigenstates as an identity. Evaluating the Time Evolution operator against one of these eigenstates yields the expected eigenvalue.\342\214\251x'|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|x'\342\214\252 = 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\342\214\251x'|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|n\342\214\252\342\214\251n|x'\342\214\252=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\342\214\251n|x'\342\214\252\342\214\251x'|n\342\214\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We now integrate the expression for the propagator over all space to eliminate the position states.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(x',t;x',0)=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\342\214\251n|x'\342\214\252\342\214\251x'|n\342\214\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=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The form of this expression is vary familiar and follows the same pattern as the Partition Function. We can extend the time domain to the complex plane and, asserting that the real part of time is zero, we can make the following equality.\316\262=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LlEhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZFLUYjNiYtSSZtZnJhY0dGJDYoLUYjNiYtSSNtaUdGJDYnUSNpdEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRidGLy9GM1EnaXRhbGljRidGU0YvRjItRiM2Ji1GUDYnUScmaGJhcjtGJ0ZTL0ZXRjFGL0YyRlNGL0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zgby8lKWJldmVsbGVkR0YxRlNGL0YyRitGU0YvRjI=Let's step back for a moment to appreciate what has happened. We have calculated the Partition Function for our system using the Path Integral Formulation. Not only that, but there seems to be some equivalence between the temperature of a system and imaginary time. This also hints at a very deep relationship between the transition probability amplitudes of a quantum system and the distribution of states in an ensemble of those systems at a fixed temperature. Ideas like this are prominent in Quantum Field Theory which uses imaginary time, more often referred to as Euclidean time, to translate back and forth between different metrics. In doing so, it is demostrated that Statistical Mechanics and Quantum Field Theory are related by Wick Rotations.We will engage in a similar practice and consider taking the time-energy Fourier Transform of the Propagator integrated over space. We note that this integral is not very well-behaved when E is a purely real argument, so we evaluate this integral with a small imaginary component added to E. The offset then dissappears when we take its vanishing 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Thus, we find that the Fourier Transform is a function which gives the energy spectrum of the system as its singular points.IV. ApplicationIV.I. Harmonic OscillatorsThe nature of the path integral makes it difficult to apply to many systems since it requires an infinite number of integrals to be evaluated. Though this is the case, we will make use of the path integral to calculate the energy spectrum of the harmonic oscillator, a very simple system which as amenable to this methodology. The harmonic oscillator is a good approximation for the bonds between atoms in diatomic molecules, but becomes more innaccurate as the anhamonicity of the real bond becomes more important at higher energies. We will use maple to calculate the spring constant of the dihydrogen system and then use that along with the Lagrangian for the Harmonic Oscillator. Using the method outlined above, we will then calculate a theoretical spectrum using the Path Integral formalism.Digits := 15:
with(QuantumChemistry):
bonds := map(x -> x/3 +.8, [fsolve(expand(ChebyshevT(10,x)))]):
molecules := map(R -> [["H",0,0,0],["H",0,0,R]], bonds):
energies := map(Energy,molecules,method='FullCI',basis="cc-PVDZ"):
pes := interp(bonds,energies,R):
eq := diff(pes,R):
R_eq := fsolve(eq, R=bonds[1] ..bonds[-1]):
plot(pes(R), R=bonds[1]..bonds[-1]):
d_pes := diff(pes, R$2):
unit_factor := (Units[Unit]('hartree'))/(Units[Unit]('angstrom')^2):
k0 := subs(R=R_eq, d_pes)*unit_factor:
k := convert(k0,units,'J/m^2');
m := AtomicData("H")[atomicweight]:
mu:= convert(m^2/(2*m),units,'kg'):
omega := simplify(sqrt(k/mu));From this, we see that the the spring constant of the H2 bond is is around 570 J/m2 with a characteristic angular frequency of 8\303\2271014 s-1. The first step in calculating the energy spectrum requires the construction of the Propagator for this system which in turn requires the Lagrangian. This is a well known result and is given below.\342\204\222=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Plugging this into the Feynman Path Integral gives the following expression for the Propagator. Note that we are considering the case that we start at some bond length x and allow the system to evolve for a time t at which point we determine the probabilty amplitude of finding the bond length to be x'.K(x,t;x,0)=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We now expand the Path Integral into its full glory by subdividing the interval into N pieces and considering the case that we take this mesh to be infinitely fine. Once again, the initial and final states are understood to be the first and last states of the index.K(x,T;x,0)=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 one could imagine, performing the operations necessary to simplify this expression into something useful is no small task. With this in mind, we will simply accept the result published by Feynman and Hibbs in the book Quantum Mechanics and Path Integrals. The classical counterpart of the Propagator for this system was actually found before the invention of Quantum Mechanics by Gustav Mehler as the Green's Function for the differential equation governing the Classical Harmonic Oscillator. In general, all Propagators are the Green's function corresponding to the Schr\303\266dinger Differential Operator.K(x,t;x,0)=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 now integrate this expression over all space to obtain the partition function of the system, noting that we have substituted \316\262, as before. K(x,-i\316\262\342\204\217;x,0)=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Though there are some notable complications in the form of hyperbolic trig functions, the only term which depends on the spatial coordinate is in the form of a Gaussian. This leads to a relativelty straightforward similification to the expected partition function of the Quantum Harmonic Oscillator.Z = 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2LlEhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZF=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 final step explicitly show the product of the coldness function with the energy eigenvalues of the Quantum Harmonic Oscillator, as we set out to do. Plugging in the angular frequency we previously determined will give us the energy spacings of the H2 bond up to the harmonic approximation. This general algorithm for finding energy eigenvalues of a quantum system has been applied to many quantum systems with equal effect.V. ConclusionThe Path Integral formalism of Quantum Mechanics is a powerful tool for lifting the curtain on the dynamics of quantum systems and the quantum origin of the Least Action Principle in Classical Dynamics. This formulation addresses the quandry that is trying to understand the origin of the Least Action Principle which is applied to quantum systems. We see that extremalizing the Action of a quantum system amounts to determining the dynamical path which picks up the smallest complex phase out of the ensemble of all possible paths. As we move away from this ideal path to nearby paths, their contributions are balanced out by the often extremely oscillatory nature of the phase introduced by this action. The collective summation of all of these paths, weighted equally subject only to rotations in the complex plane, results in interference among these paths, leaving only the path which minimizes the Action. In quantum systems, the action is small enough that it is comparable to the magnitude of \342\204\217 so multiple paths may contribute to the final outcome. This yields the "fuzziness" that we expect from quantum systems in the form of a slowly varying phase as we move away from the extremal dynamical path. It is exceptional that this formulation, which eliminates operators in an effort to move away from their black box action on states, bridges the gap between multiple length scales and system sizes.It cannot be understated that the success of the Feynman Path Integral Formulation is due, in large part, to its origins as a Green's Function Theory. The class of differential equations amenable to this methodology is vast and the determination of a Green's Function renders finding solutions for a particular member of the class trivializes their solution going forward. We also see that there are hints of a deeper structure in the relation that we found which maps the dynamics of a quantum system to its statistical mechanics by integrating the Propagator over all space. This, in and of itself, has significant implications in Quantum Field Theory which offers dimensional reduction to transform problems to Statistical Mechanics, providing another avenue by which solutions and properties may be ascertained.VI. Selected References(1) Feynman, R. P.; Hibbs, A. R.; Styer, D. F. Quantum Mechanics and Path Integrals: Emended Edition; 2010.(2) Sakurai, J. J.; Fu Tuan, S.; Newton, R. G. Modern Quantum Mechanics. Phys. Today 1986, 39 (7), 69\342\200\22370. https://doi.org/10.1063/1.2815083.(3) Feynman\342\200\231s Thesis - A New Approach to Quantum Theory; 2010. https://doi.org/10.1142/9789812567635.(4) MacKenzie, R. Path Integral Methods and Applications. 2000.(5) Grosche, C. An Introduction into the Feynman Path Integral. 1993.(6) LLC, R. Vibrational Motion and the Harmonic Oscillator. Maplesoft Online Help. https://www.maplesoft.com/support/help/maple/view.aspx?path=QuantumChemistry%2FHarmonicOscillator.Special Thanks to Andrew Dotson on YouTube for his video series on the Path Integral Formulation