Why can we always choose energy eigenstates to be purely real function

[Irishdoug]

Homework Statement

Why can we always choose energy eigenstates to be purely real functions (unlike the physical wavefunction $\psi$(x,t)?

This question is taken from an assignment on MIT's opencourses (Q2a).

https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/assignments/MIT8_04S13_ps4.pdf

Relevant Equations

N/A

I couldn't quite answer, so looked at the solution. I just want to ensure I am undertsanding the answer correctly. The answer is given here on page 3. Q2a:

https://ocw.mit.edu/courses/physics...pring-2013/assignments/MIT8_04S13_ps4_sol.pdf

Am I right in concluding that the reason energy eigenstates can be taken to be purely real is because the energy operator is purely real? And with the physical wavefunction this is not possible due to it having a momentum and as such a momentum operator that has a complex form?

Thankyou for your help!

 

[PeroK]

Homework Statement:: Why can we always choose energy eigenstates to be purely real functions (unlike the physical wavefunction **\psi**(x,t)?

This question is taken from an assignment on MIT's opencourses (Q2a).

https://ocw.mit.edu/courses/physics...-i-spring-2013/assignments/MIT8_04S13_ps4.pdf
Homework Equations:: N/A

I couldn't quite answer, so looked at the solution. I just want to ensure I am undertsanding the answer correctly. The answer is given here on page 3. Q2a:

https://ocw.mit.edu/courses/physics...pring-2013/assignments/MIT8_04S13_ps4_sol.pdf

Am I right in concluding that the reason energy eigenstates can be taken to be purely real is because the energy operator is purely real? And with the physical wavefunction this is not possible due to it having a momentum and as such a momentum operator that has a complex form?

Thankyou for your help!

The eigenstates form a basis. And any wavefunction is some complex linear combination of eigenstates. If you find a basis with complex functions you can simply rejig the basis functions so that your basis has real valued functions.

You could also rejig things so the basis functions are purely imaginary valued.

This has no physical significance.

 

[PeroK]

PS it only relies on the linearity of the Schrödinger equation and can be done for any Hermitian operator.