# Time independent wave function vs the initial wave function

• iScience
In summary, there is often confusion about the term "time independent wave function." It is typically used to describe either the Schrodinger equation or Hamiltonian. The time-independent Schrodinger equation is an Eigenvalue equation for the Hamiltonian operator. When the Hamiltonian is time-independent, the solutions of this equation, along with the initial state and a known propagator, can determine the wave function at any future point in time.

#### iScience

I thought there was a difference, but I'm being told now that the time independent wave function is the wave function at t=0

Are you sure you mean time independent wave function? The words 'the time independent' in my experience are usually used to preface either 'Schrodinger equation' or Hamiltonian. The time-independent Schrodinger equation is just the Eigenvalue equation for the Hamiltonian operator: ##H|E>=E|E>## and where the Hamiltonian is time-independent, the solutions of this, together with the initial state expressed in terms of those eigenkets, and an easily-identified propagator, are all that is needed to give the wave function at any future point in time.

## 1. What is the difference between time independent wave function and initial wave function?

The initial wave function describes the state of a quantum system at a specific point in time, while the time independent wave function describes the probability distribution of the system's state over time.

## 2. Can the initial wave function change over time?

No, the initial wave function is constant and does not change over time. It is a fixed representation of the state of the system at a specific moment.

## 3. How does the time independent wave function relate to the Schrödinger equation?

The time independent wave function is a solution to the Schrödinger equation, which describes the evolution of a quantum system over time. It allows us to calculate the probability of finding the system in a particular state at a given time.

## 4. Is the time independent wave function always the same for a given system?

No, the time independent wave function can vary depending on the potential energy of the system. Different potential energy functions can result in different time independent wave functions for the same initial wave function.

## 5. Can the time independent wave function be used to predict the exact state of a system at a specific time?

No, the time independent wave function only provides a probability distribution for the system's state over time. It cannot predict the exact state of the system at a specific time, but it can give us the probability of finding the system in a particular state.