2.4 Fourier analysis and transformations

Fourier analysis is a powerful tool in quantum mechanics, allowing us to represent functions as sums of simpler waves. It's like breaking down complex music into individual notes. This technique helps us understand how particles behave in different spaces.

By using Fourier transforms, we can switch between position and momentum representations of quantum states. This connects to the uncertainty principle and wave-particle duality, key concepts in quantum mechanics. It's a mathematical bridge between different ways of describing quantum systems.

Fourier Series Representation

Periodic Function Representation

Top images from around the web for Periodic Function Representation

• 

  Periodic summation - Wikipedia View original

  Is this image relevant?

• 

  Fourier series - Wikipedia View original

  Is this image relevant?

• 

  Expanding a periodic function into a Fourier series - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Periodic summation - Wikipedia View original

  Is this image relevant?

• 

  Fourier series - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Periodic Function Representation

• 

  Periodic summation - Wikipedia View original

  Is this image relevant?

• 

  Fourier series - Wikipedia View original

  Is this image relevant?

• 

  Expanding a periodic function into a Fourier series - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Periodic summation - Wikipedia View original

  Is this image relevant?

• 

  Fourier series - Wikipedia View original

  Is this image relevant?

1 of 3

• Fourier series represent periodic functions as an infinite sum of sines and cosines with the general form $f(x) = a_0/2 + \sum_{n=1}^{\infty} (a_n \cos(n\pi x/L) + b_n \sin(n\pi x/L))$
• Fourier coefficients $a_n$ and $b_n$ are calculated using integrals involving the function $f(x)$ multiplied by $\cos(n\pi x/L)$ or $\sin(n\pi x/L)$ respectively over one period
• Even functions have $b_n = 0$ for all $n$ (cosine series), while odd functions have $a_n = 0$ for all $n$ (sine series)
• Half-range expansions can be used for functions with half-wave symmetry (even extension) or antisymmetry (odd extension)

Convergence and Gibbs Phenomenon

• Dirichlet conditions specify sufficient conditions for a function to have a Fourier series representation the function must be absolutely integrable, have a finite number of discontinuities, and have a finite number of maxima and minima in any finite interval
• Pointwise convergence of a Fourier series means the series converges to the function value at each point where the function is continuous, while uniform convergence means the maximum error between the function and partial sums goes to zero as $n \to \infty$
  • Continuous functions with piecewise continuous derivatives have uniformly convergent Fourier series, while discontinuous but piecewise continuous functions have pointwise convergent Fourier series
  • Gibbs phenomenon describes how a Fourier series overshoots near a jump discontinuity (square wave), with the overshoot not disappearing as more terms are added

Fourier Transforms for Position and Momentum

Fourier Transform Definition and Properties

• The Fourier transform maps a function $f(x)$ to a function $F(k)$ via the integral $F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ikx} dx$, while the inverse Fourier transform maps $F(k)$ back to $f(x)$
• Properties of Fourier transforms include linearity, scaling, shifting, modulation, differentiation, and convolution
  • The convolution theorem states that convolution in position space corresponds to multiplication in momentum space and vice versa
• Parseval's theorem states that the total power of a signal is conserved under the Fourier transform, which corresponds to conservation of total probability for normalized wavefunctions

Uncertainty Principle

• The uncertainty principle can be derived using properties of the Fourier transform and states that the product of the standard deviations of position and momentum is always greater than or equal to $\hbar/2$
• This principle reflects the fundamental limit on simultaneously localizing a particle in both position and momentum spaces

Fourier Transforms in Quantum Mechanics

Wavefunctions in Position and Momentum Space

• In quantum mechanics, the Fourier transform relates the position-space wavefunction $\psi(x)$ to the momentum-space wavefunction $\phi(p)$ via $\phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-ipx/\hbar} dx$
• In the position representation, the squared modulus $|\psi(x)|^2$ represents the probability density for measuring the particle's position at $x$, while in the momentum representation, $|\phi(p)|^2$ represents the probability density for measuring the particle's momentum as $p$
• The Fourier transform relationship between $\psi(x)$ and $\phi(p)$ reflects the wave-particle duality of quantum states localized particles have broad momentum distributions and vice versa

Momentum Eigenstates and Hamiltonian

• Momentum eigenstates have wavefunctions of the form $\psi(x) = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar}$ in the position representation, and their Fourier transforms are delta functions $\delta(p-p_0)$ in the momentum representation
• For a potential $V(x)$ in the Hamiltonian, the energy eigenstates in the position representation satisfy the time-independent Schrödinger equation, while their momentum-space wavefunctions satisfy an integral equation involving the Fourier-transformed potential

Solving Differential Equations with Fourier Methods

Ordinary Differential Equations (ODEs)

• Fourier series can be used to solve linear ODEs with periodic boundary conditions by assuming the solution is an infinite series and substituting it into the ODE to derive equations for the coefficients
• Fourier sine series or cosine series are used to solve ODEs with Dirichlet (fixed) or Neumann (derivative) boundary conditions respectively on a finite interval $[0, L]$, solving the associated eigenvalue problem

Partial Differential Equations (PDEs)

• Fourier transforms can be used to solve linear PDEs on an infinite domain by transforming the PDE into an ODE in the Fourier domain, solving for the transformed solution, and applying the inverse Fourier transform to obtain the solution in position space
  • The heat equation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$ on an infinite line has solution $U(k, t) = e^{-\alpha k^2 t} F(k)$, where $F(k)$ is the Fourier transform of the initial condition $f(x) = u(x, 0)$
  • The Schrödinger equation $i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x)\psi$ for a free particle has plane-wave solutions $\psi(x, t) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \phi(p) e^{i(px-Et)/\hbar} dp$, where $E = p^2/(2m)$
• Fourier transform methods can also be used for solving PDEs on finite domains with appropriate transforms for the boundary conditions (Fourier sine and cosine transforms)