Final answer:
The valid wavefunctions for an electron from the given options are (b) Ψ = e⁻²ˣ and (c) Ψ = sin(x), as their squares result in real, non-negative values and are normalizable over all space.
Step-by-step explanation:
A wavefunction, represented by the Greek letter psi (Ψ), describes the state of an electron in quantum mechanics. A valid wavefunction must meet certain criteria, one of which is that the square of its magnitude (|Ψ|^2) represents the probability density of finding an electron at a given point in space. This probability density must be a real, non-negative quantity because probabilities cannot be negative.
The question asks which of the following represents a possible wavefunction for an electron, given that the square of the wavefunction is always a real quantity and proportional to the probability of finding an electron at a given point:
- (a) Ψ = 3x + 2
- (b) Ψ = e⁻²ˣ
- (c) Ψ = sin(x)
- (d) Ψ = x²
Each of the functions provided must be assessed on the basis that when squared, the result should be a real, non-negative quantity. Let's analyze each option:
- (a) 3x + 2 can result in negative values depending on the value of x.
- (b) e⁻²ˣ, when squared, gives a real, positive number for all values of x.
- (c) sin(x), when squared, gives real, non-negative values in the range of 0 to 1.
- (d) x² when squared, gives a real, positive number for all values of x.
Thus, the possible wavefunctions for an electron are options (b), (c), and (d) as their squares always result in real, non-negative quantities. Since the wavefunction must be normalizable (integral over all space of |Ψ|^2 must be finite), we can also rule out (a) and (d) as they are not normalizable over all space. Hence, the correct answer is the combinations of (b) and (c).