Final answer:
The function f(r), representing the potential energy of an electron in a hydrogen atom, can assume values from negative infinity up to and including zero, expressed as the interval f(r) ∈ (-∞, 0].
Correct option is c)
Step-by-step explanation:
The question asks for the possible values that a function f(r) can assume, given that it is described as the potential energy function of an electron in a hydrogen atom. According to the provided context in Equation 8.1, when r equals zero, the potential energy U(r) is negative infinity, and as r approaches positive infinity, U(r) approaches zero. Therefore, the correct interval expressing the possible values for f(r) in this context is:
f(r) ∈ (-∞, 0]
This indicates that f(r) can take on any value from negative infinity up to and including zero, but no positive values.
The quantum model of the hydrogen atom is also referenced, highlighting that due to the probability density at r = 0 being zero, the electron avoids having an infinite negative potential energy. This physical constraint on r does not affect the mathematics of the interval since it is still theoretically considered in the r → 0 limit.