02 · Objects

Basis基底

Change the axes, the numbers change, the arrow does not. $∣0 ⟩$ and $∣1 ⟩$ are one basis. $∣ + ⟩$ and $∣ - ⟩$ are another.

In English

The grid you measure against can rotate. The arrow stays put; the numbers change. Quantum |0⟩ and |1⟩ are one grid. |+⟩ and |−⟩ are the same space, different grid.

Try this: Rotate the axes. The arrow should not move. The coordinates should.
Keep this: Change the basis, change the numbers, not the object.

Lab · rotate the axes

Basis angle 22.91831180523293°

Standard: [1.5, 0.9]
In this basis: [1.73, 0.24]

A basis is a named pair of axes. Coordinates are “how much of each axis.” Change the axes, keep the arrow: the names update, the geometry does not.

|0⟩,|1⟩ versus |+⟩,|−⟩ is a 45° rotation of the question you ask, not a change of the state. Same lab, complex length-1 vectors.