Quantum mechanics introduces wave functions, mathematical tools that describe particles' behavior in the microscopic world. These functions reveal the probability of finding a particle at a specific location, connecting abstract math to physical reality.

Wave functions are essential for understanding quantum phenomena like the double-slit experiment. By calculating probability densities and distributions, we can predict particle behavior and make sense of the strange, probabilistic nature of the quantum realm.

Wave function interpretation

Physical meaning and Born interpretation

• A wave function, denoted as $\Psi(x, t)$, is a complex-valued function that describes the quantum state of a particle in space and time
• The physical interpretation of a wave function is provided by the Born interpretation, which states that the probability of finding a particle at a given location is proportional to the square of the absolute value of the wave function at that location
• The Born interpretation connects the abstract mathematical concept of a wave function to the physical reality of a particle's position
• Example: In the double-slit experiment, the wave function describes the probability of a particle passing through each slit and interfering on the screen

Probability density and its relation to wave function

• The probability density, denoted as $|\Psi(x, t)|^2$, is the square of the absolute value of the wave function and represents the probability of finding the particle per unit length at a given location and time
• The wave function itself does not have a direct physical meaning, but its square modulus (probability density) provides information about the probability distribution of the particle's position
• Example: For a particle in a box, the probability density is highest at the center of the box and zero at the walls, indicating that the particle is most likely to be found in the middle of the box

Probability density and distribution

Calculating probability density

• The probability density, $|\Psi(x, t)|^2$, is calculated by taking the square of the absolute value of the wave function at a given location and time
• To find the probability of a particle being within a specific region, integrate the probability density over that region
• Example: To determine the probability of finding an electron in the first half of a one-dimensional box, integrate the probability density from 0 to L/2, where L is the length of the box

Probability distribution and its properties

• The probability distribution, $P(x)$, is obtained by integrating the probability density over all space, which gives the probability of finding the particle at different locations
• The probability distribution is always non-negative and normalized, meaning that the total probability of finding the particle somewhere in space is equal to one
• Example: For a particle in a harmonic oscillator potential, the probability distribution is a Gaussian function centered at the equilibrium position, with the width determined by the oscillator's frequency and the particle's mass

Normalization of wave functions

Purpose of normalization

• Normalization is the process of scaling a wave function so that the total probability of finding the particle is equal to one
• Normalized wave functions are essential for maintaining the probabilistic interpretation of quantum mechanics and ensuring that the probabilities of different outcomes add up to one
• Example: In a two-state system (e.g., spin-1/2 particles), the normalized wave function ensures that the probabilities of measuring spin-up and spin-down add up to one

Procedure for normalizing wave functions

• To normalize a wave function, determine the normalization constant by integrating the square of the absolute value of the wave function over all space and setting it equal to one
• Multiply the wave function by the normalization constant to obtain the normalized wave function, ensuring that the total probability is conserved
• Example: For a Gaussian wave function $\Psi(x) = Ae^{-x^2/2\sigma^2}$, the normalization constant is $A = (1/\sqrt{2\pi\sigma^2})^{1/2}$, which ensures that the integral of $|\Psi(x)|^2$ over all space equals one

Expectation values from wave functions

Definition and calculation of expectation values

• The expectation value of an observable is the average value of that observable over many measurements on identically prepared systems
• To calculate the expectation value of an observable, multiply the observable's operator by the wave function, take the complex conjugate of the result, and integrate the product of the complex conjugate and the original wave function over all space
• Example: The expectation value of energy for a particle in a potential $V(x)$ is calculated using the Hamiltonian operator $\hat{H} = -(\hbar^2/2m)(d^2/dx^2) + V(x)$ in the expectation value integral

Expectation values of position and momentum

• The expectation value of position, $\langle x \rangle$, is calculated using the position operator, $x$, in the expectation value integral
• The expectation value of momentum, $\langle p \rangle$, is calculated using the momentum operator, $-i\hbar(d/dx)$, in the expectation value integral, where $\hbar$ is the reduced Planck's constant
• Expectation values provide information about the average behavior of a quantum system and are essential for making predictions and comparing theoretical results with experimental observations
• Example: For a particle in a harmonic oscillator potential, the expectation value of position is zero (at the equilibrium position), while the expectation value of momentum is also zero (due to the symmetric nature of the potential)