Quantum Harmonic Oscillator: First Eigenfunction and Wavefunction Analysis

The given expression is:

'ψ_1(x) = √(2/a)sin(πx/a)'

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 'π' is a mathematical constant.

The function 'sin(πx/a)' is the sine function, which oscillates between -1 and 1 as 'x' varies. Multiplying it by '√(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.