psi_1x = sqrtfrac2asinleftfracpi xaright

The given expression is:

[\psi_1(x) = \sqrt{\frac{2}{a}}\sin\left(\frac{\pi x}{a}\right)]

This is the first eigenfunction of a one-dimensional quantum harmonic oscillator. It represents the spatial wavefunction of the system.

In this expression, (x) represents the position coordinate, (a) is a constant representing the width of the potential well, and (\pi) is a mathematical constant.

The function (\sin\left(\frac{\pi x}{a}\right)) is the sine function, which oscillates between -1 and 1 as (x) varies. Multiplying it by (\sqrt{\frac{2}{a}}) scales the amplitude of the oscillation.

Overall, the expression represents a standing wave pattern, with nodes at regular intervals determined by the value of (a). The wavefunction describes the probability distribution of finding a particle in a particular position within the potential well.