How to get wavelength of the wave function?

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SUMMARY

The wavelength of the wave function for a particle in the infinite square well potential at the 3rd excited state is calculated using the formula wavelength = 2a/integer. In this case, the integer for the 3rd excited state is 3, leading to a wavelength of 2a/3. The correct answer is therefore 2/3 a, confirming the faculty's response. The wave function is represented as a sine function that approaches zero at the boundaries of the well, reinforcing the need for integral half-wavelengths within the length a.

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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 ?
 
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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.
 

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