Final Answer:
The value of 'a' is determined as 'a = π / λ', where λ represents the wavelength of the particle's wave function.
Step-by-step explanation:
The wave function Ψ(x) = A sin(ax) represents a particle's behavior in space through a sinusoidal function. Here, 'a' is related to the wavelength of the wave function. The general form of a sinusoidal wave is Ψ(x) = A sin(kx), where 'k' represents the wave number. In this specific case, comparing the given function to the general form reveals that 'k' corresponds to 'a'.
The wave number 'k' is related to the wavelength 'λ' through the equation k = 2π / λ. By equating 'k' to 'a', we find that a = π / λ. This relationship indicates that the value of 'a' in the given wave function is inversely proportional to the wavelength of the particle. A smaller 'a' value implies a longer wavelength, signifying a broader spatial distribution for the particle. Conversely, a larger 'a' value corresponds to a shorter wavelength, suggesting a more confined or localized behavior in space.
Here is complete question;
"A particle of mass m has the wave function Ψ(x) = A sin(ax) where A and a are positive real constants. Find the value of 'a' in terms of the given parameters and describe how this wave function represents the particle's behavior in space."