1.2 Quantum mechanics principles and postulates

Quantum Mechanics Postulates

Fundamental Postulates and Their Implications

Quantum mechanics rests on a small set of postulates that govern how particles behave at atomic and molecular scales. These postulates replace the deterministic rules of classical mechanics with a probabilistic framework.

Postulate 1: The Wave Function. The state of a quantum system is completely described by its wave function, $\Psi(x, t)$. This single mathematical object contains all the information you can ever extract about the system.

Postulate 2: Observables as Operators. Physical quantities like position, momentum, and energy are represented by mathematical operators that act on the wave function. The eigenvalues of these operators are the only possible outcomes you can get from a measurement. For example, the momentum operator in one dimension is $\hat{p} = -i\hbar \frac{\partial}{\partial x}$, and its eigenvalues are the allowed momentum values.

Postulate 3: Measurement and Collapse. When you measure a physical quantity, the wave function collapses into one of the eigenstates of the corresponding operator. The probability of landing in a particular eigenstate is given by the square of the absolute value of the overlap between the wave function and that eigenstate. Before measurement, the system can be in a combination of eigenstates; after measurement, it's in exactly one.

Schrödinger Equation and Heisenberg Uncertainty Principle

The Schrödinger equation governs how the wave function evolves in time. In its time-dependent form:

$i\hbar \frac{\partial}{\partial t}\Psi(x,t) = \hat{H}\Psi(x,t)$

where $\hat{H}$ is the Hamiltonian operator (total energy). This equation plays the same role in quantum mechanics that Newton's second law plays in classical mechanics: given the current state and the forces involved, it tells you what happens next.

The Heisenberg uncertainty principle places a hard limit on how precisely you can simultaneously know certain pairs of quantities. For position and momentum:

$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$

This isn't a limitation of your measuring equipment. It's built into nature. The reason traces back to the fact that the position and momentum operators don't commute: $[\hat{x}, \hat{p}] = i\hbar$. Non-commuting operators can never have simultaneously well-defined values, which is a direct consequence of the wave-like nature of quantum objects.

Wave Function and Quantum States

Properties of Wave Functions

The wave function $\Psi(x, t)$ is a complex-valued function, meaning it has both real and imaginary parts. You can't measure it directly, but its modulus squared, $|\Psi(x, t)|^2$, gives the probability density of finding the particle at position $x$ at time $t$.

A valid wave function must satisfy the normalization condition:

$\int_{-\infty}^{\infty} |\Psi(x, t)|^2 \, dx = 1$

This just says the particle has to be somewhere with 100% certainty. Wave functions must also be continuous and (typically) have continuous first derivatives, so that the physics they describe stays well-behaved.

Superposition and Entanglement

Superposition means a quantum system can exist in a combination of multiple states at once. If $\Psi_1$ and $\Psi_2$ are both valid states, then $\Psi = c_1\Psi_1 + c_2\Psi_2$ is also a valid state, where $c_1$ and $c_2$ are complex coefficients. A spin-1/2 particle, for instance, can be in a superposition of spin-up and spin-down until you measure it. The probabilities of each outcome are $|c_1|^2$ and $|c_2|^2$.

Entanglement is a correlation between particles that has no classical analog. When two particles are entangled, measuring one instantly constrains what you'll find when you measure the other, regardless of the distance between them. In a pair of entangled photons, for example, measuring the polarization of one photon immediately determines the polarization of the other. The key point: this correlation is stronger than anything classical probability can produce, as confirmed by violations of Bell's inequalities.

Solving the Schrödinger Equation

Time-Independent Schrödinger Equation

For systems where the potential doesn't change with time, you can separate out the time dependence and work with the time-independent Schrödinger equation:

$\hat{H}\Psi = E\Psi$

This is an eigenvalue equation. The Hamiltonian operator $\hat{H}$ consists of two parts:

• Kinetic energy operator: $-\frac{\hbar^2}{2m}\nabla^2$
• Potential energy operator: $V(x)$

So the full equation in one dimension reads:

$-\frac{\hbar^2}{2m}\frac{d^2\Psi}{dx^2} + V(x)\Psi = E\Psi$

To solve it, you apply boundary conditions (what happens to $\Psi$ at the edges of the system) and the normalization requirement. These constraints are what force the energy to take on only specific, discrete values.

Quantized Energy Levels and Simple Quantum Systems

The solutions to the Schrödinger equation yield quantized energy levels: the system can only have certain discrete energies, not a continuous range. This quantization is what gives quantum mechanics its name.

Three model systems show up constantly in molecular physics:

• Particle in a box: A particle confined between rigid walls (infinite potential at the boundaries). The allowed energies are $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$, where $n = 1, 2, 3, \ldots$ and $L$ is the box length. This model illustrates why confinement leads to discrete energy levels.
• Harmonic oscillator: A particle in a parabolic potential ($V(x) = \frac{1}{2}m\omega^2 x^2$). The energy levels are evenly spaced: $E_n = \hbar\omega(n + \frac{1}{2})$. The $\frac{1}{2}\hbar\omega$ ground-state energy (zero-point energy) means the particle is never completely at rest.
• Hydrogen atom: A proton-electron system solved in spherical coordinates. The energy levels go as $E_n = -13.6 \text{ eV}/n^2$. This is the simplest real atom and the starting point for understanding multi-electron systems and molecular bonding.

Quantum Mechanics vs Classical Determinism

Probabilistic Nature of Quantum Mechanics

In classical mechanics, if you know a system's initial position and momentum exactly, you can predict its future with certainty. Quantum mechanics breaks this completely. The outcome of any single measurement is fundamentally random; you can only predict the probability distribution of results.

The Born rule formalizes this: the probability of measuring a particular value of an observable is $|\Psi(x, t)|^2$ (for position) or more generally $|c_n|^2$ for finding the system in eigenstate $n$. This isn't a statement about incomplete knowledge. Even with perfect information about the wave function, the outcome of an individual measurement remains unpredictable.

Experimental Evidence and Implications

Several landmark experiments confirm the probabilistic framework:

• Double-slit experiment: Send particles (electrons, photons) one at a time through two slits. Each particle lands at a single spot on the detector, but over many trials, an interference pattern builds up. This demonstrates that each particle's wave function passes through both slits simultaneously, and the probability distribution follows wave-like interference.
• Stern-Gerlach experiment: A beam of silver atoms passes through an inhomogeneous magnetic field and splits into discrete spots rather than a continuous smear. This shows that angular momentum is quantized and that measurement yields only specific eigenvalues, not a continuum.

These results have practical consequences beyond philosophy. The probabilistic and superposition properties of quantum systems underpin technologies like quantum computing (where qubits exploit superposition to process information) and quantum cryptography (where measurement-induced collapse provides security guarantees that classical systems cannot match).