# Finding Wavefunction with just the Hamiltonian

• metapuff
However, if you know the Hamiltonian and the basis states, you can use the Schrödinger equation to determine the time evolution of the state and potentially narrow down the possible values for ##f(k)##. In summary, without knowing ##\Psi## or having more information, it is not possible to solve for the coefficients ##f(k)## in the given wavefunction. However, the Hamiltonian and basis states can be used to potentially narrow down the possibilities for ##f(k)## through the Schrödinger equation.
metapuff
Say I have a wavefunction that's a superposition of two-particle states:
$$\Psi = \int dk ~f(k) c^{\dagger}_k c^{\dagger}_{-k} | 0 \rangle$$
Here, ##|0\rangle## is the vacuum and ##c^{\dagger}_k c^{\dagger}_{-k} | 0 \rangle## represents a pair of fermions with momenta ##k,-k##. My goal is to solve for the coefficients ##f(k)## that determine the weight of each two-particle state in the superposition. Presumably, I could do this by computing
$$f(k) = \langle c_{-k}c_k | \Psi \rangle$$
However, I don't know what ##\Psi## is. I do know the form of the Hamiltonian for this system, though. Is it possible to find ##\Psi## if I only know the Hamiltonian ##H## and the form of the basis states ##c^{\dagger}_k c^{\dagger}_{-k} | 0 \rangle##? Thanks in advance! :)

No, it's not possible without more information. There are infinite possibilities for ##f(k)## and each possibility corresponds to a different state.

## 1. Can the wavefunction be found with just the Hamiltonian?

Yes, the wavefunction can be determined using the Schrödinger equation, which is based on the Hamiltonian operator.

## 2. What is the Hamiltonian?

The Hamiltonian is an operator in quantum mechanics that represents the total energy of a system, including both kinetic and potential energy.

## 3. What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system based on its Hamiltonian.

## 4. How is the wavefunction related to the Hamiltonian?

The wavefunction is a mathematical function that describes the quantum state of a system, and it can be determined using the Schrödinger equation, which is based on the Hamiltonian.

## 5. Are there any limitations to finding the wavefunction with just the Hamiltonian?

There are some limitations, as the Schrödinger equation is only applicable to non-relativistic systems and certain assumptions must be made about the system's potential energy. In some cases, additional information may be needed to fully determine the wavefunction.

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