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I. THE RULES (POSTULATES OF QUANTUM MECHANICS)
These are the axioms. Everything else follows from them.
1. State of a system Rule: A quantum system is completely described by a state : Either a wavefunction ψ ( x , t ) Or an abstract state vector ∣ ψ ⟩ 📌 Meaning:
This state contains all possible information about measurable outcomes.
2. Probability rule (Born rule) Equation: P ( x , t )= ∣ ψ ( x , t ) ∣ 2 📌 Meaning: You do not get certainty You get probabilities ∣ ψ ∣ 2 is a probability density 3. Normalization rule Equation: ∫ ∣ ψ ( x , t ) ∣ 2 dx = 1 📌 Meaning:
The particle has to be somewhere
Total probability must equal 1
4. Observables → operators Rule: Every measurable physical quantity corresponds to an operator. Examples: Position → x^ = x
Momentum → ^^ p^ =−^ i^ ℏ^ d dx Energy → Hamiltonian operator (^) H^ 📌 Meaning:
Measurements are actions on states, not numbers you already know.
5. Measurement outcomes Rule: A measurement of an observable gives one of the operator’s eigenvalues. Eigenvalue equation: ^ A ψ = aψ 📌 Meaning: Only specific values are allowed This is why energy becomes **quantized
- State collapse (measurement postulate) Rule:** After measurement, the system collapses into the eigenstate corresponding to the measured value. 📌 Meaning: Measurement changes the system This is not classical disturbance—it’s fundamental (At intro level, this is treated mathematically, not philosophically.) 7. Time evolution rule (dynamics) Equation (central law): i ℏ ∂ ψ ∂ t = H^ ψ 📌 Meaning: This tells you how quantum states evolve in time.
This is the most important equation in QM.
II. CORE EQUATIONS YOU MUST KNOW
📌 Meaning: Total energy = kinetic + potential Drives time evolution
6. Commutators Definition: [ A^ , B^ ]= A^ B^ − B^ A^ 📌 Meaning: Tells you whether two quantities can be known simultaneously 7. Canonical commutation relation [ ^ x , ^ p ]= i ℏ 📌 Meaning:
This is the mathematical root of quantum uncertainty.
8. Heisenberg uncertainty principle Equation: Δ x Δ p ≥ ℏ 2 📌 Meaning: Not a measurement flaw A fundamental limit of nature III. SPECIAL BUT ESSENTIAL RELATIONS De Broglie wavelength λ = h p 📌 Meaning: Particles behave like waves. Planck constants
h =6.626 × 10 − 34 , ℏ = h 2 π 📌 Meaning: Sets the scale where quantum effects matter. IV. DIRAC (BRA–KET) NOTATION RULES Once you move beyond wavefunctions: State vectors ∣ ψ ⟩ Inner product ⟨ ϕ ∣ ψ ⟩ Expectation value ⟨ A ⟩ = ⟨ ψ ∣ ^ A ∣ ψ ⟩ 📌 Meaning: Same physics, cleaner math. V. WHAT TO MEMORIZE VS WHAT TO UNDERSTAND Memorize early: Schrödinger equation Momentum operator Probability rule Expectation value formula Understand deeply: Operators ↔ observables Eigenvalues ↔ measurements Why probability is fundamental Why measurement changes states
