1. Historical Development of Quantum Mechanics

The Problem: Classical physics (Rayleigh-Jeans law) predicted the "ultraviolet catastrophe"—infinite energy radiated at high frequencies:

As $\nu \to \infty$, this predicts $u \to \infty$—every blackbody should emit infinite energy!

Planck's Solution (Dec 14, 1900): In an "act of desperation," Planck proposed that energy is quantized in discrete packets:

where $h = 6.626 \times 10^{-34}$ J·s is Planck's constant—now a fundamental constant of nature.

Planck's Law: The correct blackbody spectrum:

Significance: Planck himself didn't fully appreciate the revolutionary nature of his proposal. He thought energy quantization was a mathematical trick, not a fundamental property of nature. It would take Einstein to recognize the true significance.

The Puzzle: When light shines on a metal surface, electrons are ejected. Classical wave theory predicted:

• Brighter light should eject electrons with more energy
• Any frequency should work if bright enough
• There should be a time delay before electron ejection

Experiments showed ALL of these predictions were wrong!

Einstein's Radical Proposal: Light consists of particle-like quanta (later called photons):

Photoelectric equation:

where $W$ is the work function. Electrons are ejected only if $h\nu > W$, regardless of intensity!

Why it mattered: Einstein took Planck's hypothesis seriously and extended it—light itself is quantized, not just its emission/absorption. This was so radical that even Planck initially rejected it. Einstein won the 1921 Nobel Prize for this work, not relativity!

The Mystery: Atoms emit light at discrete frequencies (spectral lines), not a continuous spectrum. Hydrogen's spectrum followed the empirical Balmer formula:

but classical physics couldn't explain why!

Bohr's Bold Postulates (1913):

1. Electrons orbit in stationary states with quantized angular momentum: $L = n\hbar$
2. No radiation in stationary states (violates classical EM!)
3. Radiation emitted/absorbed only during transitions: $h\nu = E_i - E_f$

Energy levels:

Success and failure: Perfectly explained hydrogen spectrum but failed for helium and heavier atoms. Still, it introduced the crucial concept of quantum jumps between discrete energy levels.

The Bold Idea: If light (waves) can behave like particles (photons), perhaps matter (particles) can behave like waves!

de Broglie relations:

Every particle with momentum $p$ has an associated wavelength $\lambda$!

Experimental confirmation: Davisson-Germer experiment (1927) observed electron diffraction—electrons producing interference patterns just like waves!

Einstein's endorsement: Einstein immediately recognized the importance, saying de Broglie had "lifted a corner of the great veil." This was the key insight that led Schrödinger to his wave equation.

Heisenberg's radical approach: Abandon visualizable models! Only observable quantities (spectral lines, intensities) should appear in the theory.

Key insight: Physical observables are represented by matrices that don't commute:

This non-commutativity is the mathematical origin of the uncertainty principle!

Historical note: Heisenberg developed this theory while recovering from hay fever on the island of Helgoland in June 1925. He later recalled: "It was about three o'clock at night when the final result of the calculation lay before me... I was far too excited to sleep."

Schrödinger's approach: Inspired by de Broglie, sought a wave equation for matter waves:

Time-dependent Schrödinger equation:

Time-independent (energy eigenvalue) equation:

Schrödinger immediately solved this for hydrogen and reproduced Bohr's energy levels—but now from first principles!

Unification: Schrödinger proved his wave mechanics was mathematically equivalent to Heisenberg's matrix mechanics. Two completely different approaches gave the same theory!

What is $\psi$? Schrödinger thought $|\psi|^2$ represented charge density. Born proposed the correct interpretation:

gives the probability density of finding the particle at position $x$ at time $t$.

Revolutionary implication: Quantum mechanics is inherently probabilistic—not due to ignorance, but as a fundamental feature of nature. Einstein never accepted this: "God does not play dice!" Born's interpretation won the 1954 Nobel Prize.

Certain pairs of observables cannot be simultaneously measured with arbitrary precision:

This is not a limitation of measurement technique—it's a fundamental property of quantum systems!

The challenge: Schrödinger's equation is non-relativistic. Dirac sought an equation that:

• Is first-order in time (like Schrödinger)
• Is first-order in space (for relativistic symmetry)
• Reduces to Schrödinger equation at low energies

Dirac equation:

Stunning prediction: The equation predicted antimatter! Negative energy solutions led Dirac to propose the existence of positrons (discovered by Anderson in 1932). This was quantum field theory's first triumph.