# Comparing Wave Functions: Are \psi_{1} and \psi_{2} in the Same Quantum State?

• Axiom17
In summary, the conversation discusses the determination of whether two wave functions, \psi_{1} and \psi_{1}, correspond to the same quantum state of a particle. The equations and calculations presented simplify the problem to \psi_{1}(x,y,z)=A and \psi_{2}(x,y,z)=e^{z}A. The conclusion is that the two wave functions do correspond to the same quantum state and the relationship between wave functions and quantum states is discussed.
Axiom17

## Homework Statement

To determine whether two wave functions, $\psi_{1}$ and $\psi_{1}$ correspond to the same quantum state of a particle.

## Homework Equations

Calculations (simplified):

$$\psi_{1}(x,y,z)=A$$

$$\psi_{2}(x,y,z)=e^{z}A$$

## The Attempt at a Solution

The two wave functions do correspond to the same quantum state. However I can't figure out the correct wording to explain this. At the moment I just have that "$\psi_{1}$ is equal to $\psi_{2}$ with respect to the independant variable $z$ in the term $e^{z}$".

Hopefully that's correct, or at least it makes some sense.. sure there's probably a better (more correct) way to write it though. :shy:

Axiom17 said:

## Homework Statement

To determine whether two wave functions, $\psi_{1}$ and $\psi_{1}$ correspond to the same quantum state of a particle.

## Homework Equations

Calculations (simplified):

$$\psi_{1}(x,y,z)=A$$

$$\psi_{2}(x,y,z)=e^{z}A$$

## The Attempt at a Solution

The two wave functions do correspond to the same quantum state. However I can't figure out the correct wording to explain this. At the moment I just have that "$\psi_{1}$ is equal to $\psi_{2}$ with respect to the independant variable $z$ in the term $e^{z}$".

Hopefully that's correct, or at least it makes some sense.. sure there's probably a better (more correct) way to write it though. :shy:

You should use the relation of wavefunctions to quantum states. For instance in the coordinate basis $$|\vec{x}\rangle$$, we can write

$$|\psi_1 \rangle = \int d\vec{x}~ \psi_1(\vec{x}) |\vec{x}\rangle .$$

Then one way to show that two wavefunctions describe the same state would be to show that

$$\langle \vec{x}|\psi_1 \rangle =\langle\vec{x}|\psi_2 \rangle$$

Are you sure there is $$e^z$$ and not $$e^{iz}$$ in your problem? Just checking ...

## 1. What is a wave function?

A wave function, denoted as ψ, is a mathematical description of the probability of finding a particle in a certain state or location. It contains information about the particle's position, momentum, and other physical properties.

## 2. How do you compare two wave functions?

To compare two wave functions, you can look at their mathematical equations and see if they are equivalent. You can also compare their graphs to see if they have similar shapes and characteristics.

## 3. What is a quantum state?

A quantum state refers to the set of properties that fully describe a particle, such as its position, momentum, and energy. It is represented by a wave function and can change over time according to the laws of quantum mechanics.

## 4. Can two wave functions represent the same quantum state?

Yes, two wave functions can represent the same quantum state if they are equivalent in terms of their mathematical equations and physical properties. This means that they have the same probability distribution and can be used to describe the same particle.

## 5. Why is it important to compare wave functions?

Comparing wave functions allows us to understand the similarities and differences between different quantum states. This can help us make predictions about the behavior of particles and study the fundamental principles of quantum mechanics.

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