Question

This is my first question. If I don't conform to community guidelines please comment and I will happily update the question to meet the requests of the community I am asking for help.

I am trying to understand notation in the following exposition relating to the Schrödinger equation. I am a mathematician studying Liouville quantum gravity, so I want to understand this so I can further appreciate QFT in general from a physics POV. It isn't hard for me to port over mathematical intuition to novel notation or concepts, but I do get stuck with trying to understand what things mean. The information here is taken from Srendicki.

So in basic quantum mechanics the time evolution of the system is described by the Schrödinger equation:

iℏ∂∂t|ψ,t⟩=H|ψ,t⟩.(1)

I understand that the Hamiltonian operator is the sum of the kinetic and potential energy operators. In the position basis, this equation becomes

iℏ∂∂tψ(x,t)=−ℏ22m∇2ψ(x,t)(2)

where
ψ(x,t)=⟨x|ψ,t⟩.(3)

I can see how the kinetic energy operator is given by −ℏ22m∇2
, as it is the same as p⋅p2m
. What happened to the potential energy operator in this equation?

Next, can you explain what is happening with eq. (3)? From my reading, I guess that |ψ,t⟩
represents the state of the system, i.e., is a vector of quantities that can tell us everything we want to know about the system. I also take it that bras are operators. What operator is ⟨x|
, mathematically? Can you also explain the difference between ψ(x,t)
and the ψ
in the ket? Options:


A) The potential energy operator is not explicitly represented in Equation (2) due to the specific context or assumptions made in the formulation of this equation.

B) The potential energy operator is implicitly accounted for in the overall Hamiltonian, H, but it's not explicitly separated in Equation (2) in the position basis.

C)
⟨
�
∣
⟨x∣ is the position eigenstate, representing the position space representation of the wavefunction.

D)
�
(
�
,
�
)
ψ(x,t) represents the wavefunction in position space, while
∣
�
,
�
⟩
∣ψ,t⟩ represents the state vector in the abstract (ket) notation of quantum mechanics.

Answer 1

The potential energy operator is not explicitly represented in Equation (2) due to the specific context or assumptions made in the formulation of this equation. The ⟨x| operator represents the position eigenstate and ψ(x,t) represents the wavefunction in position space. ∣ψ,t⟩ represents the state vector in the abstract notation of quantum mechanics.

Step-by-step explanation:

The potential energy operator is not explicitly represented in Equation (2) due to the specific context or assumptions made in the formulation of this equation. The equation (2) in the position basis only represents the kinetic energy operator in terms of the Laplacian operator (∇2). The potential energy operator is implicitly accounted for in the overall Hamiltonian, H, which is the sum of the kinetic and potential energy operators.

The operator ⟨x| is the position eigenstate, representing the position space representation of the wavefunction. It is used to distribute the amplitude of the wavefunction across different positions in space. On the other hand, ψ(x,t) is the wavefunction expressed in position space, giving the probability amplitude of finding the particle at a specific position x and time t. In contrast, ∣ψ,t⟩ represents the state vector in the abstract (ket) notation of quantum mechanics, which contains all the necessary information about the system, including probability amplitudes and possible measurement outcomes.