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

[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 independent 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:

 

[fzero]

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 independent 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

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

 

[arkajad]

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