Postulates of Quantum Mechanics
The fundamental framework of quantum mechanics is built upon a set of core principles, often called the postulates of quantum mechanics. These postulates provide a structured foundation to describe the behavior of physical systems at the quantum scale. Below is a detailed summary of the main postulates of quantum mechanics.
1. The State of a Quantum System
In quantum mechanics, the state of a physical system is completely described by a wave function [math]\psi[/math] (or a state vector [math]|\psi\rangle[/math]) in a complex vector space called Hilbert space. The wave function provides all accessible information about the system.
• Postulate 1:
The state of a quantum system is represented by a normalized vector [math]|\psi\rangle[/math] in a Hilbert space. In position representation, this is typically given as the wave function [math]\psi(x)[/math], where [math]x[/math] represents the position.
• Normalization Condition:
The wave function must be normalized, ensuring that the total probability of finding the particle somewhere in space is 1:
[math]\int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1[/math]
2. Observables and Operators
In quantum mechanics, measurable physical quantities (observables) like position, momentum, and energy are represented by operators that act on the state vector. Each observable has an associated Hermitian operator.
• Postulate 2:
- Every measurable quantity [math]A[/math] is represented by a Hermitian operator [math]\hat{A}[/math] in Hilbert space. The eigenvalues of [math]\hat{A}[/math] represent the possible outcomes of measuring [math]A[/math].
For example, the position [math]x[/math] and momentum [math]p[/math] operators in the position representation are:
[math]\hat{x} = x, \quad \hat{p} = -i \hbar \frac{d}{dx}[/math]
3. Measurement and Eigenvalues
Upon measurement of an observable, the only possible values that can be observed are the eigenvalues of the corresponding operator. When a measurement is made, the system’s wave function collapses to the eigenstate corresponding to the measured eigenvalue.
• Postulate 3:
- If an observable [math]A[/math] is measured, the outcome of the measurement will be one of the eigenvalues [math]a_n[/math] of the operator [math]\hat{A}[math], where: [math]\hat{A} |a_n\rangle = a_n |a_n\rangle[/math] Here, [math]|a_n\rangle[/math] is the eigenstate corresponding to the eigenvalue [math]a_n[/math] . Immediately after the measurement, the system is in the eigenstate [math]|a_n\rangle[/math] associated with the measured eigenvalue [math]a_n[/math].
4. Probability and Expectation Values
The probability of obtaining a particular eigenvalue [math]a_n[/math] when measuring [math]A[/math] in the state [math]|\psi\rangle[/math] is given by the square of the projection of [math]|\psi\rangle[/math] onto the eigenstate [math]|a_n\rangle[/math].
• Postulate 4:
- The probability [math]P(a_n)[/math] of measuring the eigenvalue [math]a_n[/math] for observable [math]A[/math] is:
[math]P(a_n) = |\langle a_n | \psi \rangle|^2[/math]
• Expectation Value:
- The expectation value of an observable [math]A[/math] in the state [math]|\psi\rangle[/math] is the average value of measurements of [math]A[/math] on an ensemble of identically prepared systems in that state:
[math]\langle A \rangle = \langle \psi | \hat{A} | \psi \rangle[/math]
In position space, this can be written as:
[math]\langle A \rangle = \int_{-\infty}^{\infty} \psi^*(x) \hat{A} \psi(x) \, dx[/math]
5. Time Evolution of Quantum States
The time evolution of a quantum state is governed by the Schrödinger equation. For a closed system with a time-independent Hamiltonian [math]\hat{H}[/math], the state [math]|\psi(t)\rangle[/math] evolves according to this equation.
• Postulate 5:
- : The time evolution of a quantum state [math]|\psi(t)\rangle[/math] is governed by the Schrödinger equation: [math]i \hbar \frac{d}{dt} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle[/math] where [math]\hat{H}[/math] is the Hamiltonian operator representing the total energy of the system.
For a wave function [math]\psi(x, t)[/math], the Schrödinger equation in position space is:
[math]i \hbar \frac{\partial}{\partial t} \psi(x, t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \psi(x, t) + V(x) \psi(x, t)[/math]
where [math]V(x)[/math] is the potential energy function.
Key Concepts
- State Representation: The state of a quantum system is represented by a normalized wave function [math]\psi(x)[/math] or state vector [math]|\psi\rangle[/math] in Hilbert space.
- Observables as Operators: Physical observables correspond to Hermitian operators.
- Hermitian operators: A Hermitian operator is an operator \[math]hat{A} that satisfies [math]\langle \phi | \hat{A} \psi \rangle = \langle \hat{A} \phi | \psi \rangle for any states [math]|\phi\rangle and [math]|\psi\rangle. In quantum mechanics, Hermitian operators are crucial because their eigenvalues, representing measurable quantities, are always real.
- Measurement Postulate: Measurements yield eigenvalues of operators, and the system collapses to the eigenstate of the measured eigenvalue.
- Probability and Expectation: The probability of a measurement outcome and the expectation value of an observable can be calculated from the wave function or state vector.
- Time Evolution: The time evolution of the state vector is governed by the Schrödinger equation, which describes how quantum states evolve deterministically with time.
These postulates form the mathematical and conceptual foundation of quantum mechanics, allowing for the description and prediction of quantum systems across a variety of physical situations.