Operator representation of p in x basis.

[qubits]

For the Schrodinger equation for a hydrogen atom, we need to write out:

p^2/2m for the electron.

If we define our basis states to be a linear discrete array of points, let's say 4 points. 0,d,2d, and 3d, where is some distance, on the order of a Bohr radius. How do I write p as an operator in that positional basis? I kind of have an idea of how to do it, but I don't want to say anything to bias your answers. I'll hear your ideas then tell you what I did.

 

[qubits]

Basically I want to write this in operator notation where I have defined the basis states.

$$\hat{H} = \frac{1}{2m} \hat{p}_{1}^2 - \frac{1}{4 \pi \epsilon_0} \frac{Z e}{\hat{r}_{1} - R}$$

 

[R0man]

In the x basis, the position operator is represented as x, which acts on a wavefunction ψ(x) to give the position of the particle. In the same way, the momentum operator p is represented as p = -iħ(d/dx), which acts on ψ(x) to give the momentum of the particle.

To represent p in the given basis of 0,d,2d, and 3d, we can use the position operator x to define the basis states as |0⟩, |d⟩, |2d⟩, and |3d⟩. Then, we can write p as an operator in this basis as follows:

p = -iħ(d/dx) = -iħ((d/d0)|0⟩ + (d/dd)|d⟩ + (d/d2d)|2d⟩ + (d/d3d)|3d⟩)

= -iħ(0|0⟩ + (1/d)|d⟩ + (2/d)|2d⟩ + (3/d)|3d⟩)

= -iħ(d/d)∑n=0^3 n|nd⟩

where |nd⟩ represents the basis state at position nd.

This representation of p in the given basis shows that the momentum operator is a linear combination of the position basis states, with coefficients proportional to the distances d. This approach can be extended to any discrete basis of points, allowing for the representation of p in different bases for different systems.