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Is |x+y> = |x> + |y> ?
Thank you.
Thank you.
Is |x+y> = |x> + |y> ?.
The stuff inside a ket is some type of a label, and properties of a label depend the particular system, and on the type of label used.
For example, consider standard notation for the energy eigenstates of a harmonic oscillator. Then,
$$\left| 5 \right> \neq \left| 2 \right> + \left| 3 \right>.$$
What am i missing?
No, if you define it that way you will only cause confusion for others.I see, so I can define |x+y> = |x> + |y>, but it's not implied automatically.
The problem with your expression has nothing to do with Quantum Mechanics specifically, but rather results from a confusion between mathematical objects, on the one hand, and labels for mathematical objects, on the other. Your expression is analogous to the following classical expression relating the velocities of two people: $$\vec {V}_{Jack} + \vec {V}_{Jill} = \vec {V}_{Jack + Jill}$$ There are situations where a family of labels can indeed have a useful algebraic structure (e.g. time ##t##), but the labels used in Dirac kets in this way are not among them.Is |x+y> = |x> + |y> ?