How to get wavelength of the wave function?

[cooper607]

A particle in the infinite square well potential is found at the 3-rd excited state. What is the
wavelength of the wave function of the particle?
this is a mcq from my faculty who denoted the correct answer to be 2/3 a ...how to get that ?

 

[gadong]

The wave function is a sine function that goes to zero at the boundaries of a region of length a.

Therefore, there must be an integral (1, 2, 3...) number of half-wavelengths in a region a.

That is, a = integer*wavelength/2, or wavelength = 2a/integer.

For the ground state, the integer = 1, for the second excited state the integer = 3. It seems that the examiner has misread the question when giving the answer.