Hilbert space
A Hilbert space is the vector space in which quantum states live. For Susskind’s early spin examples it is finite-dimensional. For particles in space it becomes an infinite-dimensional space of functions.
Plain-language picture
Think less “arrows in ordinary space” and more “objects that can be added, scaled by complex numbers, and inner-producted.” A two-component spinor and a square-integrable wavefunction are different-looking objects, but both can be vectors in the quantum sense.
What structure matters
A quantum Hilbert space needs:
- vector addition and complex scalar multiplication;
- an inner product such as ⟨A|B⟩;
- normalization, so total probability is one;
- orthogonality, so distinguishable states can be represented as perpendicular directions.
Finite versus continuous examples
For one spin:
|Ψ⟩ = α|u⟩ + β|d⟩For one particle on a line, the state can be represented in the position basis by a function ψ(x), with inner products computed by integration.
Common pitfalls
- Do not treat Hilbert space as ordinary physical space.
- Do not forget the complex numbers; phases are essential.
- Do not assume a wavefunction is a different kind of state from a ket. It is usually a representation of the ket in a basis.