8.1 Path integral formulation of quantum mechanics

Path integrals offer a fresh take on quantum mechanics, summing over all possible paths a particle can take between two points. This approach connects classical and quantum physics, making it easier to visualize complex quantum phenomena.

In this section, we'll explore how path integrals work, their advantages, and how to use them to solve quantum problems. We'll see how this powerful tool applies to everything from simple particles to complex quantum fields.

Path integrals in quantum mechanics

Overview of path integrals

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• Path integrals provide an alternative formulation of quantum mechanics equivalent to the Schrödinger equation and Heisenberg matrix mechanics
• In the path integral approach, the probability amplitude for a particle to propagate from one point to another is calculated by summing over all possible paths connecting the two points, each path weighted by a phase factor
• The path integral formulation is based on the principle of least action, which states that the path taken by a particle between two points is the one that minimizes the action
  • The action is defined as the integral of the Lagrangian along the path, where the Lagrangian is the difference between the kinetic and potential energy of the system
• Path integrals can be used to calculate transition amplitudes, expectation values of observables, and the time evolution of quantum states (e.g., calculating the probability of a particle moving from point A to point B)

Advantages of path integral formulation

• The path integral approach provides a natural way to incorporate symmetries and constraints into quantum mechanics, such as gauge invariance and topological effects
  • For example, path integrals can be used to study the quantum mechanics of particles moving on curved surfaces or in the presence of electromagnetic fields
• Path integrals allow for the treatment of systems with an infinite number of degrees of freedom, such as quantum field theories
  • This is because the path integral formulation can be generalized to include multiple particles, fermions, and fields
• The path integral formulation provides a unified framework for describing both quantum and classical mechanics
  • In the classical limit, the path integral reduces to the principle of least action, which governs the motion of classical particles

Derivation of path integrals

Deriving path integrals from the Schrödinger equation

• The path integral formulation can be derived from the Schrödinger equation by dividing the time evolution into infinitesimal time steps and inserting a complete set of position eigenstates at each step
• The transition amplitude between two position eigenstates at adjacent time steps is given by the exponential of the action multiplied by $i/\hbar$, where $\hbar$ is the reduced Planck constant
  • Mathematically, the transition amplitude between position eigenstates $|x_i\rangle$ and $|x_{i+1}\rangle$ at times $t_i$ and $t_{i+1}$ is given by: $\langle x_{i+1}|e^{-iH(t_{i+1}-t_i)/\hbar}|x_i\rangle = e^{iS[x(t)]/\hbar}$ where $H$ is the Hamiltonian and $S[x(t)]$ is the action functional
• The total transition amplitude is obtained by multiplying the amplitudes for each time step and integrating over all possible intermediate positions, resulting in a path integral

Properties of path integrals

• The path integral formulation reduces to the Schrödinger equation in the limit of infinitesimal time steps and continuous paths
  • This can be shown by expanding the exponential in the path integral to first order in the time step and taking the limit as the time step goes to zero
• The normalization factor for the path integral is chosen to ensure that the total probability is conserved and that the path integral reproduces the correct classical limit
  • The normalization factor is given by the square root of the determinant of the second derivative of the action, evaluated at the classical path
• The path integral formulation is invariant under coordinate transformations and gauge transformations, which makes it a powerful tool for studying systems with symmetries

Transition amplitudes with path integrals

Calculating transition amplitudes

• Transition amplitudes can be calculated by evaluating the path integral over all possible paths connecting the initial and final states
  • For example, the transition amplitude for a particle to propagate from position $x_i$ at time $t_i$ to position $x_f$ at time $t_f$ is given by: $\langle x_f, t_f|x_i, t_i\rangle = \int_{x(t_i)=x_i}^{x(t_f)=x_f} Dx(t) e^{iS[x(t)]/\hbar}$ where $Dx(t)$ denotes the integration over all possible paths $x(t)$ connecting the initial and final positions
• For a free particle, the path integral can be evaluated exactly using Gaussian integration, yielding the free particle propagator
  • The free particle propagator is given by: $\langle x_f, t_f|x_i, t_i\rangle = \sqrt{\frac{m}{2\pi i\hbar(t_f-t_i)}} e^{im(x_f-x_i)^2/2\hbar(t_f-t_i)}$ where $m$ is the mass of the particle

Perturbative expansion and Feynman diagrams

• In the presence of a potential, the path integral can be evaluated perturbatively by expanding the action around the classical path and treating the deviations as small fluctuations
  • The leading order term in the perturbative expansion corresponds to the classical action, while higher order terms give quantum corrections
• Feynman diagrams provide a graphical representation of the perturbative expansion, with each diagram corresponding to a specific contribution to the transition amplitude
  • For example, the first-order correction to the free particle propagator due to a potential $V(x)$ is given by the Feynman diagram:

       i
    -------
    |     |
    |  V  |
    |     |
    -------
       f
    

    which represents the integral: $-\frac{i}{\hbar}\int_{t_i}^{t_f} dt \langle x_f, t_f|V(x)|x_i, t_i\rangle$

Generalization to quantum field theory

• The path integral approach can be generalized to include multiple particles, fermions, and fields, leading to the formulation of quantum field theory
  • In quantum field theory, the path integral is performed over all possible field configurations, rather than particle paths
  • The action functional for a quantum field theory is given by the integral of the Lagrangian density over spacetime
• Path integrals provide a natural framework for studying the quantum mechanics of gauge theories, such as quantum electrodynamics and quantum chromodynamics
  • In gauge theories, the path integral must be modified to include a sum over all possible gauge configurations, in addition to the sum over field configurations

Solving quantum problems with path integrals

Quantum harmonic oscillator

• The path integral formalism can be used to solve the quantum harmonic oscillator by evaluating the path integral for a quadratic potential
  • The action functional for a harmonic oscillator with mass $m$ and frequency $\omega$ is given by: $S[x(t)] = \int_{t_i}^{t_f} dt \left[\frac{1}{2}m\dot{x}^2 - \frac{1}{2}m\omega^2x^2\right]$
  • The path integral can be evaluated exactly using Gaussian integration, yielding the propagator for the harmonic oscillator: $\langle x_f, t_f|x_i, t_i\rangle = \sqrt{\frac{m\omega}{2\pi i\hbar \sin\omega(t_f-t_i)}} e^{iS_{\text{cl}}[x(t)]/\hbar}$ where $S_{\text{cl}}[x(t)]$ is the action evaluated along the classical path

Double-slit experiment and quantum tunneling

• For the double-slit experiment, the path integral approach provides a natural explanation for the interference pattern by summing over paths that pass through each slit
  • The transition amplitude for a particle to propagate from the source to the screen is given by the sum of the amplitudes for paths passing through each slit: $\langle x_f, t_f|x_i, t_i\rangle = \langle x_f, t_f|x_1, t_1\rangle\langle x_1, t_1|x_i, t_i\rangle + \langle x_f, t_f|x_2, t_2\rangle\langle x_2, t_2|x_i, t_i\rangle$ where $x_1$ and $x_2$ are the positions of the slits and $t_1$ and $t_2$ are the times at which the particle passes through each slit
• The path integral formulation can be used to study quantum tunneling, where a particle can penetrate a potential barrier that would be classically forbidden
  • The transition amplitude for a particle to tunnel through a potential barrier is given by the path integral over all paths that cross the barrier, even those with imaginary action
  • The probability of tunneling decreases exponentially with the height and width of the barrier, as predicted by the WKB approximation

Semiclassical approximation and constrained systems

• In the semiclassical limit, the path integral can be approximated using the method of stationary phase, which selects the paths that extremize the action
  • The semiclassical approximation is valid when the action is much larger than $\hbar$, which corresponds to the limit of large quantum numbers or high energies
  • The semiclassical propagator is given by: $\langle x_f, t_f|x_i, t_i\rangle \approx \sum_{\text{cl}} \sqrt{\frac{1}{2\pi i\hbar}\left|\frac{\partial^2 S_{\text{cl}}}{\partial x_f \partial x_i}\right|} e^{iS_{\text{cl}}[x(t)]/\hbar}$ where the sum is over all classical paths connecting the initial and final positions
• The path integral approach can be applied to study the quantum mechanics of systems with constraints, such as the motion of a particle on a curved surface or the dynamics of a spinning top
  • Constraints can be incorporated into the path integral by introducing Lagrange multipliers and modifying the action functional accordingly
  • For example, the path integral for a particle constrained to move on a sphere of radius $R$ is given by: $\int Dx(t) D\lambda(t) e^{iS[x(t),\lambda(t)]/\hbar}$ where $\lambda(t)$ is a Lagrange multiplier enforcing the constraint $|x(t)|=R$

Many-body systems and quantum field theory

• Path integrals provide a powerful tool for studying the quantum mechanics of many-body systems, such as the interacting electron gas or the Bose-Einstein condensate
  • The path integral for a many-body system involves a sum over all possible configurations of the particles, taking into account their interactions and quantum statistics
  • The path integral formulation can be used to derive effective field theories for many-body systems, such as the Ginzburg-Landau theory of superconductivity or the Gross-Pitaevskii equation for Bose-Einstein condensates
• In quantum field theory, path integrals are used to calculate correlation functions and scattering amplitudes
  • The path integral for a quantum field theory involves a sum over all possible field configurations, weighted by the exponential of the action functional
  • Feynman diagrams provide a systematic way to calculate perturbative corrections to the correlation functions and scattering amplitudes
  • The path integral formulation is essential for the study of non-perturbative effects in quantum field theory, such as instantons and solitons