Susskind QM Lecture 2 - Quantum States

One-line takeaway

A quantum state is a vector of amplitudes, and probabilities come from squared magnitudes of those amplitudes.

Plain-language map

Lecture 2 turns the spin experiments into vector language. The working assumption is operational: after a complete preparation, there is no further accessible hidden detail that would let us predict every incompatible measurement. For a single spin, the state space is two-dimensional.

Choose the z-basis |u⟩, |d⟩. Any spin state can be written as a superposition of those basis vectors:

|A⟩ = α_u |u⟩ + α_d |d⟩

The coefficients are amplitudes. They are not probabilities; they are complex numbers whose magnitudes squared give probabilities for the corresponding z-measurement outcomes.

P(up)   = |α_u|²
P(down) = |α_d|²
|α_u|² + |α_d|² = 1

The same physical state can be expanded in another basis, such as the x-basis |r⟩, |l⟩. Changing basis changes the components but not the underlying state.

Core ideas

• Quantum state as complete predictive information.
• Hilbert space as the vector space where states live.
• Bra-ket notation for amplitudes such as ⟨u|A⟩.
• Measurement and state preparation as the bridge from vector to outcome.

Common pitfalls

• Do not square the coefficient itself if it is complex; use squared magnitude.
• Do not treat a basis expansion as a physical mixture. A superposition has relative phases that can matter later.
• Do not confuse “two-dimensional state space” with ordinary two-dimensional physical space.

Links

• Susskind QM Lecture 1 - Systems and Experiments
• Susskind QM Lecture 3 - Principles of Quantum Mechanics
• Susskind Quantum Mechanics Equations and definitions