Any quick help with rearranging schrodinger equation in dirac notation

In summary, the speaker is studying relativistic corrections and perturbation theory using hydrogen and is having trouble understanding the rearrangement of an equation involving kinetic energy and coulomb potential. They notice a potential error with the substitution and question why the coulomb potential is positive when it should be negative for an attractive potential.
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I'm looking through my lecture notes, (studying relativistic corrections/perturbation theory using hydrogen), and I seem to have a mind block with one of the equations (the last one from the 3 in the middle).

I know that the kinetic energy and coulomb potential has been subbed in for the operator H0, but I don't understand the rearrangement. Simply multiply both sides by 2*m, then take the coulomb potential to the other side.

The coulomb potential (the e^2 term) on the left hand side is positive, moving it to the right would make it negative, so why is it postive?
 

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I know that the kinetic energy and coulomb potential has been subbed in for the operator H0, but I don't understand the rearrangement. Simply multiply both sides by 2*m, then take the coulomb potential to the other side.
... I think I see what you mean.
##V(r)=e^2/4\pi\epsilon_0r## would be a repulsive potential wouldn't it?
If the potential is supposed to be attractive, then they've made a typo in the initial substitution: there should be a minus sign in there.
 

1. How do I write the Schrodinger equation in Dirac notation?

The Schrodinger equation in Dirac notation is written as iℏ∂/∂t|Ψ(t)⟩ = Ĥ|Ψ(t)⟩, where i is the imaginary unit, ℏ is the reduced Planck's constant, ∂/∂t is the partial derivative with respect to time, |Ψ(t)⟩ is the state vector, and Ĥ is the Hamiltonian operator.

2. How do I rearrange the Schrodinger equation in Dirac notation to solve for the state vector?

To rearrange the Schrodinger equation in Dirac notation to solve for the state vector, you can simply divide both sides of the equation by iℏ∂/∂t, resulting in |Ψ(t)⟩ = (1/iℏ)Ĥ|Ψ(t)⟩. This will allow you to solve for the state vector |Ψ(t)⟩ using the Hamiltonian operator Ĥ.

3. Can I use Dirac notation to represent any quantum state?

Yes, Dirac notation can be used to represent any quantum state. The state vector |Ψ⟩ represents the state of a system at a particular point in time, and it can be used to represent any quantum state, whether it is a single particle or a multi-particle system.

4. How do I convert a Schrodinger equation in position space to Dirac notation?

To convert a Schrodinger equation in position space to Dirac notation, you can use the position operator x and momentum operator p to write the Hamiltonian operator as Ĥ = p^2/2m + V(x). Then, you can substitute this into the Schrodinger equation in Dirac notation and solve for the state vector |Ψ(t)⟩.

5. What is the significance of the Dirac notation in quantum mechanics?

The Dirac notation is a powerful tool in quantum mechanics as it allows for a concise and elegant representation of quantum states and operations. It simplifies calculations and allows for easier manipulation of equations, making it a fundamental tool in understanding and solving problems in quantum mechanics.

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