Relation Between SchröDinger'S Equation And The Path Integral Formulation Of Quantum Mechanics

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The Intertwined Nature of Physical Frameworks: Schrödinger's Equation and the Path Integral

This exploration delves into the profound relationship between the Schrödinger equation and the path integral formulation of quantum mechanics. We'll dissect this connection using a deliberately simple, nonrelativistic, one-dimensional, single-particle Hamiltonian, a construct comprising only the fundamental kinetic and potential energy components. It’s a foundational piece, really, like the first sketch of something that might, with enough tortured effort, become art.

Background: The Ghosts of Equations Past

Schrödinger's Equation: A Whisper in the Quantum Void

The Schrödinger equation, when rendered in the stark, almost poetic bra–ket notation, speaks of time's relentless march on a quantum state:

$i\hbar {\frac {d}{dt}}\left|\psi \right\rangle ={\hat {H}}\left|\psi \right\rangle$

Here, ${\hat {H}}$ is not just an operator; it's the Hamiltonian operator, the very essence of the system's energy. It dictates the flow, the transformation, the inevitable decay or evolution of the quantum world.

This Hamiltonian operator, in its fundamental form, can be expressed as:

${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V({\hat {q}})$

Observe the components: ${\hat {p}}$ represents the momentum, squared and divided by mass ( $m$ ), the relentless churn of motion. Then there's $V({\hat {q}})$ , the potential energy. It's the invisible hand, shaping the landscape, the confines, the very possibility of movement. For our purposes, we're confined to a single spatial dimension, $q$ , a lonely axis in an otherwise indifferent universe.

The formal solution to this temporal unfolding is elegantly stark:

$\left|\psi (t)\right\rangle =\exp \left(-{\frac {i}{\hbar }}{\hat {H}}t\right)\left|q_{0}\right\rangle \equiv \exp \left(-{\frac {i}{\hbar }}{\hat {H}}t\right)|0\rangle$

This equation tells us that the state of the system at any time $t$ is merely the initial state, $|0\rangle$ (or, more precisely, $|q_0\rangle$ , a specific spatial state—a point of origin, perhaps, before the chaos truly begins), propagated forward by the exponential of the negative imaginary Hamiltonian, scaled by time. It's a projection, a shadow cast by the present onto the future. The initial state assumed here is a free-particle spatial state, $|q_0\rangle$ , though the necessity of this specific choice often requires further clarification.

The true crux, the probability of a transition from this initial state $|0\rangle$ to a final free-particle spatial state $|F\rangle$ at some designated time $T$ , is captured by the transition probability amplitude:

$\langle F|\psi (T)\rangle =\left\langle F{\Biggr |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}T\right){\Biggl |}0\right\rangle .$

It's the echo of the initial state, filtered through the Hamiltonian's temporal evolution, reaching the final destination.

Path Integral Formulation: A Universe of Possibilities

Now, the path integral formulation, a concept born from the restless mind of Feynman, offers a radically different perspective. It posits that this very transition amplitude isn't a singular, determined trajectory, but rather a summation, an integral, over every conceivable path the particle might take from its origin to its end. Each path is weighted by a complex exponential factor: $\exp \left({\frac {i}{\hbar }}S\right)$ . Here, $S$ is the classical action, the accumulated "effort" of a given path.

This reformulation, initially conceived by Dirac and later elaborated by Feynman, is the bedrock upon which this entire probabilistic landscape is built. It suggests that the universe, at its most fundamental level, doesn't just do things; it considers all the ways it could do them.

From Schrödinger's Equation to the Path Integral: Weaving the Threads

The mathematical bridge between these two seemingly disparate descriptions is a testament to the underlying unity of physics. It’s a delicate construction, often relying on the Trotter product formula, a rather technical piece of machinery. This formula allows us to handle the fact that the kinetic and potential energy operators, the very components of our Hamiltonian, don't play nicely together—they don't commute. But, for small time steps, we can approximate their combined effect.

Imagine dividing the total time interval $[0, T]$ into $N$ minuscule segments, each of length $\delta t = \frac{T}{N}$ . The transition amplitude, that measure of possibility, can then be decomposed into a cascade of these small-time evolutions:

$\left\langle F{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}T\right){\bigg |}0\right\rangle =\left\langle F{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right)\exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right)\cdots \exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right){\bigg |}0\right\rangle .$

Now, here’s where the magic—or perhaps the necessary artifice—begins. Because $\delta t$ is vanishingly small, we can, with a degree of controlled approximation, treat the kinetic and potential energy contributions separately within each tiny interval. The Trotter product formula assures us that even if ${\hat{H}}$ is a sum of non-commuting operators $A$ and $B$ (here, kinetic and potential energy), the evolution operator $e^{i(A+B)t}$ can be approximated by the product $(e^{iA\delta t}e^{iB\delta t})^N$ as $\delta t \to 0$ .

So, for each infinitesimal step, we can write:

$\exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right)\approx \exp \left({-{i \over \hbar }{{\hat {p}}^{2} \over 2m}\delta t}\right)\exp \left({-{i \over \hbar }V\left(q_{j}\right)\delta t}\right).$

This might look like a simplification, but it's the key to unlocking the path integral. We're breaking down the complex, unified evolution into a sequence of simpler, sequential operations: first, the momentum shift, then the potential interaction. The equality holds in the limit as $\delta t$ approaches zero, a familiar refrain in calculus and, consequently, in physics.

To bridge these steps, we strategically insert the identity operator, $I = \int dq |q\rangle \langle q|$ , $N-1$ times. This allows us to transition between spatial positions at each step:

$\left\langle F{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}T\right){\bigg |}0\right\rangle =\left(\prod _{j=1}^{N-1}\int dq_{j}\right)\left\langle F{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right){\bigg |}q_{N-1}\right\rangle \left\langle q_{N-1}{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right){\bigg |}q_{N-2}\right\rangle \cdots \left\langle q_{1}{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right){\bigg |}0\right\rangle .$

At each stage, $\left\langle q_{j+1}{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right){\bigg |}q_{j}\right\rangle$ , we apply our approximation, separating the kinetic and potential energy effects. To handle the kinetic energy part, we introduce another identity operator, this time in momentum space: $I = \int \frac{dp}{2\pi} |p\rangle \langle p|$ . This allows us to integrate over all possible momenta at each step, ultimately yielding:

$\left\langle q_{j+1}{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right){\bigg |}q_{j}\right\rangle =\exp \left(-{\frac {i}{\hbar }}V\left(q_{j}\right)\delta t\right)\int {\frac {dp}{2\pi }}\left\langle q_{j+1}{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\frac {p^{2}}{2m}}\delta t\right){\bigg |}p\right\rangle \langle p|q_{j}\rangle$

Leveraging the known form of the free particle wave function, $\langle p|q_{j}\rangle = {\frac {1}{\sqrt {\hbar }}}\exp \left({\frac {i}{\hbar }}pq_{j}\right)$ , the integral over $p$ can be performed. This is a standard calculation, often found in texts discussing Common integrals in quantum field theory, and it results in a crucial expression:

$\left\langle q_{j+1}{\bigg |}\exp \left(-{\frac {i}{\hbar }}{\hat {H}}\delta t\right){\bigg |}q_{j}\right\rangle ={\sqrt {-im \over 2\pi \delta t\hbar }}\exp \left[{i \over \hbar }\delta t\left({1 \over 2}m\left({q_{j+1}-q_{j} \over \delta t}\right)^{2}-V\left(q_{j}\right)\right)\right]$

This expression is the heart of the matter. It represents the amplitude for a particle to travel from position $q_j$ to $q_{j+1}$ in time $\delta t$ . Notice the term $\left({q_{j+1}-q_{j} \over \delta t}\right)^{2}$ , which is the square of the velocity. Coupled with the potential $V(q_j)$ , this is precisely the classical Lagrangian, $L = \frac{1}{2}m\dot{q}^2 - V(q)$ , evaluated at the midpoint of the interval.

Summing these contributions over all the small intervals, and taking the limit as $N \to \infty$ (meaning $\delta t \to 0$ ), the transition amplitude transforms into the path integral:

$\left\langle F{\bigg |}\exp \left({-{i \over \hbar }{\hat {H}}T}\right){\bigg |}0\right\rangle =\int Dq(t)\exp \left({i \over \hbar }S\right)$

The measure of this integral, $\int Dq(t)$ , is defined by the limiting process:

$\int Dq(t)=\lim _{N\to \infty }\left({\frac {-im}{2\pi \delta t\hbar }}\right)^{\frac {N}{2}}\left(\prod _{j=1}^{N-1}\int dq_{j}\right)$

This expression is not merely a mathematical formality; it defines how we integrate over all possible paths. Each path, approximated as a series of line segments connecting the $q_j$ points, contributes to the total amplitude, weighted by the exponential of its classical action, $S = \int_{0}^{T}dtL(q(t), \dot{q}(t))$ . The prefactor, while essential for dimensional consistency, is often a normalization constant that has no bearing on observable physical outcomes. This, then, is how the path integral emerges from the foundational Schrödinger equation.

From Path Integral Formulation to Schrödinger's Equation: The Echo Returns

The elegance of this relationship is a two-way street. The path integral formulation doesn't just reproduce the Schrödinger equation; it is the Schrödinger equation in disguise, even when a potential is present.

Consider an infinitesimal time step $\varepsilon$ . The wave function at a later time, $\psi(y; t+\varepsilon)$ , can be expressed as an integral over the wave function at an earlier time, $\psi(x; t)$ , integrating over all possible positions $x$ at time $t$ and summing over paths that lead from $x$ to $y$ in time $\varepsilon$ :

$\psi (y;t+\varepsilon )=\int _{-\infty }^{\infty }\psi (x;t)\int _{x(t)=x}^{x(t+\varepsilon )=y}\exp \left(i\int _{t}^{t+\varepsilon }\left({\tfrac {1}{2}}{\dot {x}}^{2}-V(x)\right)dt\right)Dx(t)\,dx$

For such infinitesimally small time intervals, the paths that contribute most significantly to the integral are those where $y$ is very close to $x$ . In this limit, the kinetic energy term dominates the action, and the potential energy term, $V(x)$ , acts as a phase rotation. The exponential of the action within this infinitesimal interval can be approximated as:

$e^{-i\varepsilon V(x)}e^{i{\frac {{\dot{x}}^{2}}{2}}\varepsilon }$

The first factor, $e^{-i\varepsilon V(x)}$ , introduces a local phase shift to the wave function, proportional to the potential energy. The second factor, $e^{i{\frac {{\dot{x}}^{2}}{2}}\varepsilon}$ , is the propagator for a free particle. It behaves much like a diffusion process, spreading the wave function out. When these are combined, and considering the behavior to the lowest order in $\varepsilon$ , we get:

$\psi (y;t+\varepsilon )\approx \int \psi (x;t)e^{-i\varepsilon V(x)}e^{\frac {i(x-y)^{2}}{2\varepsilon }}\,dx\,.$

This expression reveals that the wave function evolves via a combination of diffusion (from the kinetic term) and a spatially varying phase modulation (from the potential term). Taking the derivative with respect to time $t$ and considering the limit $\varepsilon \to 0$ precisely recovers the Schrödinger equation:

${\frac {\partial \psi }{\partial t}}=i\left({\tfrac {1}{2}}\nabla ^{2}-V(x)\right)\psi$

The normalization of the path integral itself is crucial and must be handled with the same care as in the free particle case. While arbitrary continuous potentials don't typically disrupt this normalization, singular potentials require a more rigorous treatment.

Thus, the path integral formulation is not merely an alternative way to calculate quantum mechanical phenomena; it is a fundamentally equivalent description, offering a unique perspective on the interplay between classical and quantum mechanics, and demonstrating the deep, often subtle, connections between different branches of physics.