2.3 Probability Density and Expectation Values

Quantum mechanics gets weird when we zoom in on tiny particles. Probability density and wave functions help us make sense of this bizarre world. They tell us where particles are likely to be and how they behave.

Expectation values are like quantum averages. They bridge the gap between math and real-world measurements. By calculating these values, we can predict what we'll observe when we study quantum systems.

Probability Density and Wave Functions

Defining Probability Density

• Probability density measures the likelihood of locating a particle at a specific spatial position
• Calculated by squaring the absolute value of the wave function, |Ψ(x,t)|^2
• Integrating the probability density over a region of space yields the probability of finding the particle within that region
• The total probability of finding the particle anywhere in space must equal 1, ensured by normalizing the wave function

Wave Function Characteristics

• The wave function, Ψ(x,t), is a complex-valued function describing the quantum state of a particle
• Contains comprehensive information about the particle's behavior and properties (position, momentum, energy)
• The complex conjugate of the wave function, Ψ*(x,t), is used in expectation value calculations
• Normalizing the wave function ensures the total probability of finding the particle somewhere is 100%

Expectation Values of Observables

Calculating Expectation Values

• Expectation values represent the average value of an observable over multiple measurements on identically prepared systems
• The general formula for calculating the expectation value of an observable A is A = ∫Ψ*(x,t)ÂΨ(x,t)dx, where  is the operator corresponding to the observable A
• Position expectation value, x, is calculated using the position operator x̂: x = ∫Ψ*(x,t)xΨ(x,t)dx
• Momentum expectation value, p, is calculated using the momentum operator p̂ = -iħ(d/dx): p = ∫Ψ*(x,t)(-iħ(d/dx))Ψ(x,t)dx
• Energy expectation value, E, is calculated using the Hamiltonian operator Ĥ: E = ∫Ψ*(x,t)ĤΨ(x,t)dx

Operators for Observable Quantities

• Each observable quantity (position, momentum, energy) has a corresponding operator
• The position operator, x̂, multiplies the wave function by x
• The momentum operator, p̂ = -iħ(d/dx), involves the derivative of the wave function with respect to position
• The Hamiltonian operator, Ĥ, represents the total energy of the system and is used to calculate the energy expectation value

Physical Meaning of Expectation Values

Interpreting Expectation Values

• Expectation values provide the average value of an observable if measured repeatedly on identically prepared systems
• The position expectation value, x, represents the average position of the particle, weighted by the probability density
• The momentum expectation value, p, represents the average momentum of the particle, weighted by the probability density
• The energy expectation value, E, represents the average energy of the particle, weighted by the probability density

Connecting Mathematics to Physical Quantities

• Expectation values bridge the gap between the abstract mathematical description (wave function) and measurable physical quantities
• They allow for the extraction of meaningful physical information from the wave function
• Expectation values help predict the most likely outcomes of measurements on quantum systems
• Comparing expectation values with experimental results helps validate quantum mechanical models

Probability Density in Quantum Systems

One-Dimensional Infinite Square Well

• Probability density is highest at the antinodes (peaks) of the wave function
• Probability density is zero at the nodes (points where the wave function is zero)
• The number of nodes increases with increasing energy levels (ground state has no nodes, first excited state has one node, etc.)

Harmonic Oscillator

• In the ground state, the probability density is concentrated around the equilibrium position
• For higher excited states, the probability density becomes more spread out and exhibits a greater number of peaks and nodes
• The spacing between peaks and nodes is related to the energy difference between adjacent levels

Hydrogen Atom

• The probability density for the electron depends on the principal (n), angular momentum (l), and magnetic (m) quantum numbers
• The radial probability density gives the probability of finding the electron at a certain distance from the nucleus
• Different combinations of quantum numbers lead to various orbital shapes (s, p, d, f) with distinct probability density distributions

Double-Well Potential

• The probability density can exhibit symmetric or antisymmetric distributions, depending on the energy level and the shape of the potential
• In the ground state, the probability density may be concentrated in one well or equally distributed between both wells
• Higher energy levels display more complex probability density patterns with multiple peaks and nodes

Two-Dimensional Infinite Square Well

• Probability density forms patterns that reflect the symmetry of the system (rectangular, circular, or triangular)
• High probabilities occur at the antinodes of the wave function, while nodes have zero probability
• The number and arrangement of nodes depend on the energy level and the quantum numbers describing the state