Question

Choose whether the statement is true or false
The radial wave function for the ground state of H-atom decreases exponentially towards zero as ‘r’ increases.
A. True
B. False

Answer 1

Hint: The wave function ${{\psi }}$ (shi) is used to define different properties of electron e.g., energy, position, and velocity. The equation of this wave function has two parts radial function and angular function. The radial function depends upon the ‘r’ component of the polar coordinates (i.e., r, ${{\theta , }}\phi $).

Complete Step by step answer: The radial wave function as mentioned in the hint depends on the ‘r’ component of the polar coordinates, the ‘r’ represents the distance of the orbital from the nucleus.
The mathematical expression for the wave function ${{\psi }}$ can be written as;
${\psi _{(r,\theta ,\varphi )}} = {R_{(r)}}{Y_{(\theta ,\varphi )}}$
Where ${R_{(r)}}$ is the radial wavefunction and ${Y_{(\theta ,\varphi )}}$ is the angular wavefunction,
The radial wavefunction is dependent on ‘r’ and can be expressed as,
${R_{(r)}} = Np(r){e^{ - kr}}$
Where,
N is positive normalizing constant
$p(r)$ is a polynomial in ‘r’
k is a positive constant
Since the exponential component is always positive, the sign ${R_{(r)}}$ depends on the behavior of the polynomial of ‘r’ . Because the exponential has a negative sign in the exponent (${e^{ - kr}}$) the value of ${R_{(r)}}$ will approach zero(0) as the distance ‘r’ approaches infinity.
Therefore the statement, that the radial wave function for the ground state of H-atom decreases exponentially towards zero as ‘r’ increases. Is indeed true.

Hence, the correct answer is option (A)i.e., True

Note: There are different equations for the wave function, to determine different factors, such as the equation for determining the energy is, $\mathop {\text{H}}\limits^ \wedge {{\psi = E\psi }}$ where $\mathop {{H}}\limits^ \wedge $ is a mathematical operator and E is energy.
The equation to find the probability of the position of the electron in the free space is ${\text{|}}{{\text{\psi }}^2}{\text{|}}