Principles of quantum mechanics

Quantum mechanics is a beautiful and highly successful theory that describes the physical properties of nature at the smallest scales, typically the atomic and subatomic levels. It serves as a foundation for many parts of modern physics and has numerous applications in science and technology. Despite its complexity, the core of quantum mechanics can be narrowed down to a few fundamental principles. Let's understand these principles in depth, outline the main principles and illustrate them through examples and visual representations.

Wave function

The first principle of quantum mechanics is that all information about a quantum system is contained in the wave function, often denoted as ψ (psi). The wave function is a mathematical object that encodes the probabilities of all possible outcomes of a quantum measurement.

The wave function itself does not represent the physical state in the way that classical systems do. Instead, it represents a probability amplitude, where the probability of a system being in a particular state is determined by the modulus squared of the wave function.

P(x) = |ψ(x)|²

Here, P(x) is the probability density of finding the particle at position x, and ψ(x) is the wave function of the position.

Visual example: wave function

The above curve displays an example of a wave function, ψ(x). The oscillations suggest variations in probabilities, where some regions have higher or lower probability densities.

Operators and observables

In quantum mechanics, physical quantities are not measured directly as in classical mechanics. Instead, they are associated with operators. The second postulate states that every physical observable, such as position, momentum, or energy, is represented by a linear Hermitian operator. An observable O is represented by an operator Ô.

Ôψ = oψ

When a measurement is made, the system collapses to an eigenstate of the operator, and the observed value o is the eigenvalue associated with that eigenstate.

Example: measurement of position

Suppose we have an observable such as a position x, represented as an operator X̂. When we measure the position of a particle, the wave function collapses to an eigenstate of X̂, and the outcome is one of the eigenvalues.

Quantization and eigenvalues

The measurement postulate implies that various measured physical quantities are quantized. The eigenvalues of operators represent the possible outcomes that can be measured for an observable. This quantization arises naturally from the solution of the equation of operators.

For example, consider an electron in a hydrogen atom. The energy levels obtained by solving the Schrödinger equation give discrete eigenvalues corresponding to quantized energy levels.

Ĥψ = Eψ

Here, E symbolizes the energy eigenvalue, and Ĥ is the Hamiltonian operator.

Visual example: quantized energy levels

It represents the energy levels in an atom, where the vertical bars represent the quantized energy states available to an electron within the atom.

Time evolution of a system

Quantum mechanics dictates that the evolution of the state of a quantum system over time is governed by the Schrödinger equation. The wave function evolves deterministically according to this wave equation. The time-dependent Schrödinger equation for a non-relativistic quantum system is given as:

iħ ∂ψ/∂t = Ĥψ

where i is the imaginary unit, ħ is the reduced Planck constant, t is time, and Ĥ is the Hamiltonian operator describing the total energy of the system.

Theory of measurement

Measurement theory posits that performing a measurement on a quantum system causes the system to 'collapse' into an eigenstate of the observable being measured. This concept addresses the transition from quantum possibilities (described by the wave function) to a single definite outcome observed experimentally.

For example, if we measure the position of a particle, the wave function "compresses" into one of the possible point eigenstates of the position operator.

Probability explanation

The probability interpretation, or Born rule, tells us that the probability of finding a system in a particular state is related to the square of the amplitude of its wave function.

P(a) = |c(a)|²

where c(a) is the probability amplitude. This principle underlies why the probability density is calculated as the square of the magnitude of the wave function.

Text example: probability density

Consider a particle that is forced to move in a one-dimensional ring. The wave function of the particle can be:

ψ(x) = A e^(ikx)

To find the probability density function for detecting the particle at any position x, we evaluate |ψ(x)|²:

P(x) = |A|²

The probability remains constant around the ring, indicating a uniform distribution and emphasizing the uncertainty inherent in quantum mechanics.

Superposition principle

One of the phenomenal principles of quantum mechanics is the superposition principle, which states that if a system can be in several different states, then the system can also be in all of these states simultaneously. Linear combinations of these states form new valid states. Mathematically, this means:

ψ = c₁ψ₁ + c₂ψ₂ + ... + cₙψₙ

where c₁, c₂, ..., cₙ are complex numbers and ψ₁, ψ₂, ..., ψₙ are states.

Visual example: superposition of states

The red and blue lines represent the individual states ψ₁ and ψ₂ (ψ₁), respectively. The purple line represents the superposition ψ, which connects these states. Here, superposition allows the quantum system to explore a much richer set of states than is classically possible.

The role of spin

Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei. It has no classical analogue, although it can be represented mathematically in the same way as regular angular momentum.

For particles such as the electron, spin is quantized and described by the quantum number s. Electrons have a spin number of 1/2, resulting in two possible spin states, "spin-up" and "spin-down".

Example: spin and measure

Suppose a particle with spin-1/2 is in a superposition of spin-up |↑⟩ and spin-down |↓⟩:

ψ = a|↑⟩ + b|↓⟩

Upon measurement, the system collapses to spin-up or spin-down with probabilities |a|² and |b|², respectively.

Conclusion

The principles of quantum mechanics provide a framework for understanding the behaviour of systems at the quantum level, going beyond the deterministic world of classical physics into a realm governed by probability and strange, less-explained phenomena such as superposition and entanglement. This framework has not only enhanced our understanding of the atomic world but has also led to significant technological advances. Despite the complexity introduced by quantum mechanics, it remains an essential aspect of physics, continually inspiring new ideas, interpretations and advances.

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