Braket notation question

[DuckAmuck]

Is |x+y> = |x> + |y> ?

Thank you.

 

[A. Neumaier]

No. Take the inner product with $<x|$ to see that if $y$ is nonzero, the left size vanishes but the right side diverges.

 

[George Jones]

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,

$$\left| 5 \right> \neq \left| 2 \right> + \left| 3 \right>.$$

 

[DuckAmuck]

<x + y|x + y> = ∫ (x+y)^2 dV = ∫ (x^2 + y^2 + x*y + y*x) dV

(<x| + <y|)(|x> + |y>) = <x|x> + <y|y> + <x|y> + <y|x> = ∫ (x^2 + y^2 + x*y + y*x) dV

What am i missing?

 

[DuckAmuck]

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,

$$\left| 5 \right> \neq \left| 2 \right> + \left| 3 \right>.$$

I see, so I can define |x+y> = |x> + |y>, but it's not implied automatically.

 

[PeterDonis]

What am i missing?

The definition of what the x and y labels mean, i.e., what Hilbert space you are talking about and how the labels x and y index vectors in that Hilbert space. Without that you have no way of writing down integrals for the inner products at all, let alone knowing that they take the exact form you gave in post #4.

 

[A. Neumaier]

I see, so I can define |x+y> = |x> + |y>, but it's not implied automatically.

No, if you define it that way you will only cause confusion for others.

 

[Physics Footnotes]

Is |x+y> = |x> + |y> ?

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.