Path Integral Formulation of Quantum Mechanics

Da Teng, University of Chicago

There are many mutually equivalent physical formulations of quantum mechanics. The two earliest and most well known are Schrodinger's wavefunction formulation and Heisenberg's matrix formulation. After that, there were attempts to generalize classical mechanics concepts into quantum mechanics, which saw the introduction of Bohm formulation and Wigner formulation, etc. However, it was until Richard P. Feynman, that a rigid connection between quantum mechanics and one of the most important quantity in classical mechanics, the action, was established. According to himself, he was immediately fascinated by the idea of path integral when his high school physics teacher introduced it to him. While it is not that incredible that this genius finally connected the path integral with quantum mechanics, it is still astonishing how beautiful this theory can be.

Motivation

One of the key issues in quantum mechanics is the time evolution of a given system. Given a particular state, or wavefuctionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiNRIUYnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLyUrZXhlY3V0YWJsZUdGPi9GM1Enbm9ybWFsRic=at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5, we would like to know the motion of the system in following time frames. Technically, we want to 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From the formal solution of the time-dependent Schrodinger's equation, and expanding it into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== space, we can write

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

All the information required for evaluating the time evolution is in the first term in the integrand, or the matrix element of the time propagator 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. Mathematically, it really resembles the density matrix in statistical mechanics. With the definition]

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, or 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

the time evolution matrix will be exactly the same as the Boltzmann density matrix. However, in quantum mechanics, the kinetic energy operator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== and the potential energy operator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== do not commute. Hence,

To overcome this, we need the Trotter's theorem to help break the two terms apart. Trotter's theorem states that, when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== are both close enough to zero, the effect of this non-commutation will be negligible. Mathematically, it means

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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1JI21pR0YkNiVRIlBGJy8lJ2l0YWxpY0dGQi9GM1EnaXRhbGljRictRi82LVEtJnJpZ2h0YXJyb3c7RidGMi9GNlEmZmFsc2VGJy9GOUZZL0Y7RkIvRj1GWS9GP0ZZL0ZBRlkvRkRGWS9GRlEsMC4yNzc3Nzc4ZW1GJy9GSUZbby1GLzYtUSgmaW5maW47RidGMkZYRlovRjtGWUZmbkZnbkZobkZpbkZFL0ZJRkdGMi8lLGFjY2VudHVuZGVyR0ZZLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYoLUYjNiYtRmVvNiUtRi82L1EvJkV4cG9uZW50aWFsRTtGJy8lK2V4ZWN1dGFibGVHRkIvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjJGNUY4RjpGPEY+L0ZBRjdGQ0ZFL0ZJUSwwLjExMTExMTFlbUYnLUYjNigtRi82L1EqJnVtaW51czA7RidGYXBGY3BGMkZYRlpGYG9GZm5GZ25GaG5GaW4vRkZRLDAuMjIyMjIyMmVtRicvRklGX3EtRk42J1EoJiM5NDY7VUYnRlFGYXBGY3BGUy1GLzYvUSIvRidGYXBGY3BGMkZYRlpGZW5GZm5GZ25GaG5GaW4vRkZGSkZILUkjbW5HRiQ2JlEiMkYnRmFwRmNwRjItRk42J0ZQRlFGYXBGY3BGU0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GZW82JUZecC1GIzYnRltxLUZONidRKCYjOTQ2O1RGJ0ZRRmFwRmNwRlNGZHFGXHJGMkZeckZccEYyRmFwRmNwRjIvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRiM2Jy1GTjYnRlBGUUZhcC9GZHBRKDJEfk1hdGhGJ0ZTRlFGYXBGYnNGU0Zeci1GLzYvUSJ+RidGYXBGYnNGMkZYRlpGYG9GZm5GZ25GaG5GaW5GRUZhby1GLzYvUSgmZXF1aXY7RidGYXBGYnNGMkZYRlpGYG9GZm5GZ25GaG5GaW5Gam5GXG8tRiw2JS1GLzYvRjFGYXBGYnNGMkY1RjhGOkY8Rj5GQEZDRkVGSC1GIzYmRmBzLUYvNi9GV0ZhcEZic0YyRlhGWkZlbkZmbkZnbkZobkZpbkZqbkZcby1GLzYvRl9vRmFwRmJzRjJGWEZaRmBvRmZuRmduRmhuRmluRkVGYW9GMkZiby1GZW82JS1GTjYnUScmIzkzNztGJy9GUkZZRmFwRmNwRjItRiM2JEZgc0YyRl5yLUYvNi9RIi5GJ0ZhcEZic0YyRlhGWkZgb0ZmbkZnbkZobkZpbkZFRmFvRjI=

Then, the matrix element can be written as

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 =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

The only job left is to evaluate the matrix elements for the infinitesimal time evolution operator.

Connection with Path Integral

Before going into the math, we need to look into the structure of the last integral. The integral goes over an infinite number of intermediate "check points" between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=, which defines a trajectory between these two points uniquely. The integrand is the amplitude of this trajectory. This is like a summation over all possible path between two specific ends. Recall that in the Lagrangian formulation of classical mechanics, the real path between two fix end points is the path with the smallest, largest, or invariant action. This analogy will be the key for us to connect the path integral with this quantum time propagator.

We then evaluate the individual bra-kets here. The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==-th bra-ket will be

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

The potential energy part can be evaluated directly, then we have

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

Then, the kinetic energy part can be evaluated in the momentum space

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 = 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

We know the relationship that

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

Then the integral comes to be

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 = 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

Technically, this is a Fresnel integral, but we can treat it like a Gaussian integral, which yields

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

Putting everything together, and substitute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRjFGMg== back with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEjaXRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIvRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRkwtRiw2JVEoJiM4NDYzO0YnL0YwRj1GOS8lK2V4ZWN1dGFibGVHRj1GOQ==, the integral then becomes

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 = 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

If we denote the infinitesimal time 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, we can rewrite the whole thing as

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 = 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

Interestingly, the sum in the exponential, although not tractable to directly evaluate in most non-trivial cases, is somehow a Riemann sum which can be expressed as an integral. When the partition LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ==is large, and therefore LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiMxMDEzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxRjI= goes very small, the sum is just

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 = 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

We can immediately identify that, the integrand in the bracket is the Lagrangian of the system. Hence, the integral is the classical action of a given path from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=! The amplitude is therefore

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 = 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

This integral can be interpreted physically. For each trajectory from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=, we can compute the classical action LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYmLUYsNiVRInhGJ0YvRjItRjY2JC1GIzYlLUYsNiVRInRGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGR0ZERkdGRy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZERkc= as a functional of the trajectory LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInRGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA=, and the total amplitude of this transition is the sum over the contribution from all these paths, each at a phase 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The integral over all paths, is usually designated as a functional integral, which is

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

Till here, we established the essential concepts of path integral quantum mechanics. Looking into the matrix elements of a propagator, we found this amplitude is related with the classical action. For convenience, we designate this matrix element as kernel, that is,

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

Stationary Phase Approximation

Stationary phase approximation, or saddle point approximation, is an approximation to deal with integrals with forms like this.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZELUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR0Y5L0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUZINiVRInhGJ0ZLRk1GL0YvLUklbXN1cEdGJDYlLUYsNi1RLyZFeHBvbmVudGlhbEU7RidGLy9GM1EmdW5zZXRGJy9GNkZobi9GOEZobi9GO0Zobi9GPUZobi9GP0Zobi9GQUZobkZCL0ZGUSwwLjExMTExMTFlbUYnLUYjNistRkg2I1EhRictRiM2JUZjby1JI21uR0YkNiRRImlGJ0YvRi8tRkg2JVEiZkYnRktGTS1GUDYkLUYjNihGVEZLLyUrZm9yZWdyb3VuZEdRK1swLDE2MCw4MF1GJy8lLHBsYWNlaG9sZGVyR0Y5LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y5Rk1GL0ZLRmNwRmZwRmhwRk0vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYsNi1RIn5GJ0YvRjJGNS9GOEY0RjovRj1GNEY+RkBGQkZFLUYsNi1RMCZEaWZmZXJlbnRpYWxEO0YnRi9GZ25GaW5Gam5GW29GXG9GXW9GXm9GQkZFRlRGLw==

Here, we assume LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1GLDYjUSFGJ0Y9RkA=and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1GLDYjUSFGJ0Y9RkA= to be defined on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjFGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIn5GJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWS8lK2V4ZWN1dGFibGVHRjRGNQ==to simplify things, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA= has one and only one stationary point. We can see later that the assumption can be lifted such that there can be multiple stationary points. We know that the exponential part of the integrand is an oscillating function, and when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA= gets large, it oscillates very fast.

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We found out that the at large LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRj0vJSVvcGVuR1EifGdyRicvJSZjbG9zZUdGQUY6Rj0=, the positive part and the negative part of the function cancels very quickly. As long as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA= is "smoother," these regions don't really contribute to the whole integral. You can try different integration limits to verify this. We can also see that as the phase factor is divided by \342\204\217, the oscillation will be even faster, and the integral will vanishes in regions really close to the classical trajectory.

plot((x + x^2*cos(x)/2)*cos(x^4), x = -4 .. 5);

integrate((x + x^2*cos(x)/2)*cos(x^4), x=2.5...4.2);

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JCIrKj5iXkkjISM4

Stationary phase approximation states that, only the regions close to the stationary point of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA= have large contribution. Therefore, we can expand the "phase function" LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA= to its stationary phase. That is,

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

where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21uR0YkNiVRIjBGJ0Y1L0Y5USdub3JtYWxGJ0ZBLyUvc3Vic2NyaXB0c2hpZnRHRkAtRi82I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0ZB is the stationary point. Then, the integral approximately becomes

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYmUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JS1GIzYlLUklbXN1YkdGJDYlLUYsNiZRInhGJ0Y0RjdGOi1GIzYkLUkjbW5HRiQ2JVEiMEYnRjcvRjtRJ25vcm1hbEYnRk4vJS9zdWJzY3JpcHRzaGlmdEdGTUYrRk5GN0ZOLUklbXN1cEdGJDYlLUkjbW9HRiQ2LlEvJkV4cG9uZW50aWFsRTtGJ0Y3Rk4vJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRmVuLyUpc3RyZXRjaHlHRmVuLyUqc3ltbWV0cmljR0Zlbi8lKGxhcmdlb3BHRmVuLyUubW92YWJsZWxpbWl0c0dGZW4vJSdhY2NlbnRHRmVuLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictRiM2Jy1GSzYlUSJpRidGN0ZOLUYsNiZRImZGJ0Y0RjdGOi1GPjYlLUYjNiRGQkZORjdGTi1GVjYuUSJ+RidGN0ZOL0ZaUSZmYWxzZUYnL0ZnbkZocC9GaW5GaHAvRltvRmhwL0Zdb0ZocC9GX29GaHAvRmFvRmhwRmJvL0Zmb0Zkb0ZOLyUxc3VwZXJzY3JpcHRzaGlmdEdGTUZORistRlY2LVErJkludGVncmFsO0YnRk5GZ3BGaXAvRmluRjZGW3EvRl1vRjZGXXFGXnFGYm9GX3FGKy1GVjYtRmZwRk5GZ3BGaXBGanBGW3FGXHFGXXFGXnFGYm9GX3EtRiw2JVEkZXhwRicvRjVGaHBGTi1GPjYkLUYjNihGKy1GIzYqLUkmbWZyYWNHRiQ2KC1GIzYmRistRiM2JUYrLUZLNiRGXHBGTkZORitGTi1GIzYkLUZLNiVRIjJGJ0Y3Rk5GTi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZnMvJSliZXZlbGxlZEdGaHBGZHBGXXAtRlY2LlEiJ0YnRjdGTkZncEZpcEZqcEZbcUZccUZdcUZecS9GY29GZ29GX3FGW3RGPS1GUzYlLUY+NiUtRiM2JkZCLUZWNi5RKCZtaW51cztGJ0Y3Rk5GZ3BGaXBGanBGW3FGXHFGXXFGXnEvRmNvUSwwLjIyMjIyMjJlbUYnL0Zmb0ZpdEZFRk5GN0ZORlxzRmBxRk5GKy8lK2V4ZWN1dGFibGVHRmhwL0Y4USVUZXh0RidGTkZORmdxLUZWNi1RMCZEaWZmZXJlbnRpYWxEO0YnRk5GWUZmbkZobkZqbkZcb0Zeb0Zgb0Zib0ZfcS1GLDYlRkdGNEY6Rlt1Rl11Rk4=

And the integral is a Gaussian integral, which gives the final answer

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

Our take home message for this approximation is, when an oscillating phase factor is changing, only regions around its minima contribute to any integral over it.

Go back to Schrodinger's equation

Our previous derivation is based on the fact that the time propagator has the form 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, which is a result from the good old Schrodinger's equation. Hence, the dynamics from the path integral paradigm should be exactly the same as in Schrodinger's paradigm. We expect we can go back to Schrodinger's equation from our expression for kernels. Before going back to the fundamental equation of everything,

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbW9HRiQ2LlEifkYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRi1JI21pR0YkNiZRJyYjOTY4O0YnLyUnaXRhbGljR0Y3Ri9GMi1JKG1mZW5jZWRHRiQ2JS1GIzYoLUZKNiZRInhGJy9GTlEldHJ1ZUYnRi8vRjNRJ2l0YWxpY0YnLUYsNi5RIixGJ0YvRjJGNS9GOUZYRjpGPEY+RkBGQkZEL0ZIUSwwLjMzMzMzMzNlbUYnLUZKNiZRInRGJ0ZXRi9GWS8lK2V4ZWN1dGFibGVHRjcvRjBRJVRleHRGJ0YyRi9GMkYrLUYsNi5RIj1GJ0YvRjJGNUY4RjpGPEY+RkBGQi9GRVEsMC4yNzc3Nzc4ZW1GJy9GSEZmb0YrLUYsNi5RKyZJbnRlZ3JhbDtGJ0YvRjJGNUY4L0Y7RlhGPC9GP0ZYRkBGQkZERkctRlA2Ji1GIzYvLUZKNiVGVkZXRlktRiw2LVEifGdyRidGMkY1RjhGW3BGPEY+RkBGQi9GRVEsMC4xMTExMTExZW1GJy9GSEZncC1JJW1zdXBHRiQ2JS1GLDYtUS8mRXhwb25lbnRpYWxFO0YnRjIvRjZRJnVuc2V0RicvRjlGYHEvRjtGYHEvRj1GYHEvRj9GYHEvRkFGYHEvRkNGYHFGREZocC1GIzYtLUYsNi1RKiZ1bWludXMwO0YnRjJGNUY4RjpGPEY+RkBGQi9GRVEsMC4yMjIyMjIyZW1GJy9GSEZdci1GSjYlUSRpSHRGJ0ZXRlktRiw2LVEiL0YnRjJGNUY4RltwRjxGPkZARkIvRkVRLDAuMTY2NjY2N2VtRicvRkhGZnItRko2JVEoJiM4NDYzO0YnRk1GMkZXLyUrZm9yZWdyb3VuZEdRK1swLDE2MCw4MF1GJ0Zeby8lLHBsYWNlaG9sZGVyR0ZYLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0ZYRmBvRlkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYsNi1GLkYyRjVGOEY6RjxGPkZARkJGREZHRmNwLUZKNiVRInlGJ0ZXRllGV0Zbc0Zeb0Zec0Zgc0Zgb0ZZRjIvJSVvcGVuR1EpJmxhbmdsZTtGJy8lJmNsb3NlR1EpJnJhbmdsZTtGJy1GUDYmLUYjNixGZ3NGY3AtRko2JUZMRk1GMkZXRltzRl5vRl5zRmBzRmBvRllGMkZqc0ZddEZlcy1GLDYtUTAmRGlmZmVyZW50aWFsRDtGJ0YyRl9xRmFxRmJxRmNxRmRxRmVxRmZxRkRGR0Znc0Zlcy1GLDYtRmRvRjJGNUY4RjpGPEY+RkBGQkZlb0Znb0Zlc0Zeb0Zgb0YyLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZELUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR0Y5L0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUZINiVRInhGJ0ZLRk0tRiw2LVEiLEYnRi9GMi9GNkY5L0Y4RjRGOi9GPUY0Rj5GQEZCL0ZGUSwwLjMzMzMzMzNlbUYnLUZINiVRInlGJ0ZLRk1GL0YvLUZINiVRJyYjOTY4O0YnL0ZMRjRGLy1GUDYkLUYjNiZGaW5GVy1JI21uR0YkNiRRIjBGJ0YvRi9GLy1GLDYtUTAmRGlmZmVyZW50aWFsRDtGJ0YvL0YzUSZ1bnNldEYnL0Y2RlxwL0Y4RlxwL0Y7RlxwL0Y9RlxwL0Y/RlxwL0ZBRlxwRkJGRUZpbkYv

we need to apply the stationary phase approximation, to make the calculation of our kernel tractable.

From the fundamental assumption of any physics, the principle of least action, we know that the real classical trajectory of a Lagrangian dynamics between two fixed points has the least action among all possible trajectories. Since our kernel is a sum over all possible paths (or more specifically, a functional integral), we can expect the stationary phase approximation to tell us, only those paths around the true path, which has the least LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==, will contribute significantly to our sum. Or, more radically under this approximation, we can write

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,

where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYmUSNjbEYnRjJGNUY4L0Y5USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiNRIUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGQA== is the classical action.

Previously, we have determined that, for an infinitesimal motion between two points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==, the kernel can be written as

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

When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ==and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== are close, we can write 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 The wavefunction for time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== will be

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

From now on we will drop the time if the term is at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5. Then we can do a Taylor series on both sides, and of anything evaluated at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== to express them as values at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

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

The leading term will tell us about the normalizing factor.

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 = 1

Then we can say the first order term will vanish because of the odd symmetry.

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 = 0

The second order term can be integrated in closed form

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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNidGKy1GIzYmRistRiM2JUYrLUkjbW5HRiQ2JFEiaUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y8RitGPC1GLDYlUS4mIzg0NjM7JiM5NDk7RicvJSdpdGFsaWNHUSV0cnVlRicvRj1RJ2l0YWxpY0YnRkJGRS1GIzYlLUYsNiVRIm1GJ0ZCRkVGQkZFLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZRLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjw=

Then the whole thing, neglecting terms higher than first order, is

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

This simplifies into

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiQtSSNtbkdGJDYlUSJpRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y6RitGOi1GLDYmUSkmaHNsYXNoO0YnLyUnaXRhbGljR1EmZmFsc2VGJ0Y3RjpGOg==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 =LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8=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

which is just the time-dependent Schrodinger's equation!

Finally, we went back with our old friend, Schrodinger's equation, when we do the stationary phase approximation with the path integral formulation. The approximation has two parts.

1. First, we did a Taylor series on the starting point and the end point. This is an easier to justify assumption because we are looking into the differentiations of our wavefunction.

2. "The classical limit." In our arguments, we approximated the kernel with the sole contribution from the classical path. But how good is this approximation? It can be reiterated that, the SPA states that when a trajectory has some positive contribution to the kernel, an infinitesimally adjacent path will provide a equal but negative phase contribution. However, around the classical trajectory where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== has its minimum, at least in first order, the action won't change. The scope where this first order stationary property relies on the magnitude of \342\204\217. We are measuring a classical function with a tiny unit \342\204\217, and that's why the phase factor will oscillate very fast. If we need to be specific, only when the change of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== is no bigger than the scale of \342\204\217, we can say the SPA is valid.

With the argument above, we can also conclude that, when we take LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoJiM4NDYzO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUS0mcmlnaHRhcnJvdztGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi1JI21uR0YkNiRRIjBGJ0YyRjI= limit, strictly only the least action trajectory will contribute to the kernel. This is the case we will go back to classical mechanics. In the path integral scheme, trajectories other than the deterministic classical one will contribute in the transition probability, which is purely a quantum effect. How many trajectory around this classical trajectory will count in is controlled by the factor, Planck constant.

Harmonic Oscillator

Harmonic oscillators are the easiest system we can deal with. From classical mechanics, we know that the action function for a harmonic oscillator is

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

where 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is the time difference between the starting and end point.

Also, in quantum mechanics, under Born-Oppenheimer approximation (although not a good one with hydrogen), the electronic potential energy surface is quite harmonic around its bond length, making the harmonic approximation fairly reasonable for small oscillations.

with(QuantumChemistry); with(Statistics);

bond_distances := [0.4, 0.5, 0.6, 0.7, 0.75, 0.8, 0.9, 1, 1.3, 2, 5];

molecules := [seq([["H", 0, 0, 0], ["H", 0, 0, R]], R in bond_distances)];

energies := [seq(Energy(molecule, method = FullCI, basis = "cc-pVDZ"), molecule in molecules)];

ScatterPlot(bond_distances,energies);JSFH

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NiI=

From the calculation with Full-CI above, which gives the exact dissociation curve under the given basis set, is non-harmonic at very small LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== and large LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==. However, the PES is still harmonic around its optimal bond length.

Given the action function, the kernel of such a harmonic oscillator will be

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

We can expand this in terms of 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With Euler's equation, we know that the exponential can be expanded in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkZXhwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tRiw2JVEqbmkmIzk2OTtURicvRjBRJXRydWVGJy9GM1EnaXRhbGljRicvJStleGVjdXRhYmxlR0YxRjJGMkZYRjI=, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== is an integer. The pre-factor will give a 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, the series will be in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEkZXhwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYnLUkjbW9HRiQ2LVEqJnVtaW51czA7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tRjY2JC1GIzYnLUYsNiVRIm5GJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1GOzYtUSIrRidGMkY+RkBGQkZERkZGSEZKRkxGTy1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGMi1GIzYlLUZdbzYkUSIyRidGMkZYRlovJS5saW5ldGhpY2tuZXNzR0Zfby8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zpby8lKWJldmVsbGVkR0YxLyUrZXhlY3V0YWJsZUdGMUYyRjItRiw2JVEpaSYjOTY5O1RGJ0ZYRlpGXnBGMkYyLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUY7Ni1RIixGJ0YyRj4vRkFGWUZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBRLDAuMzMzMzMzM2VtRictRjs2LVEifkYnRjJGPkZARkJGREZGRkhGSkZdcS9GUEZecUZecEYygiving that the energies (which is the spectrum of the kernel), will be

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

which agrees with the answer from directly solving the eigenvalue problem.

Reference

Richard P. Feynman, Albert R. Hibbs, Daniel F. Styer, Quantum Mechanics and Path Integrals, New York: Dover

Mark Tuckerman, Statistical Mechanics: Theory and Molecular Simulation, Oxford Graduate Text