CALC (a) What is the probability that an electron in the 1s...

CALC (a) What is the probability that an electron in the 1s state of a hydrogen atom will be found at a distance less than $a / 2$ from the nucleus? (b) Use the results of part (a) and of Example 41.4 to calculate the probability that the electron will be found at distances between $a / 2$ and $a$ from the nucleus.

Key Concepts

Normalization of the Wavefunction

Normalization ensures that the total probability of finding the electron anywhere in space is equal to one. Solving problems involving electron probability in atoms requires that the wavefunction is properly normalized, which means performing an integral over all space and setting it equal to one. This principle underlies all probability calculations in quantum mechanics.

Integration in Spherical Coordinates

Due to the spherical symmetry of the hydrogen atom's 1s wavefunction, calculations of probability involve integration using spherical coordinates. The integration takes into account not only the radial dependence of the wavefunction but also the angular distribution, which simplifies considerably for spherically symmetric states. This approach is essential for accurately determining probabilities in atomic systems.

Bohr Radius

The Bohr radius, often denoted by a, is a fundamental constant that sets the scale for the size of the hydrogen atom in its lowest energy state. It appears in the analytical form of the hydrogen atom’s wavefunctions and influences the spatial distribution of the electron probability. Calculations of probabilities within fractions of the Bohr radius help illustrate the radial concentration of the electron in the 1s state.

Quantum Probability Density

In quantum mechanics, the probability density is given by the square of the absolute value of the wavefunction. This density represents the likelihood of finding a particle at a particular point in space. For an electron in an atom, it is used to calculate the probability of the electron’s presence within a specified region by integrating the probability density over that region.

Radial Probability Distribution

For spherically symmetric systems like the hydrogen atom's 1s state, the probability of finding the electron at a distance between r and r + dr is obtained by multiplying the probability density by the volume element in spherical coordinates, typically expressed as 4?r² dr. This radial probability distribution emphasizes the role of both the wavefunction's decay and the increasing volume at larger radii, making it a crucial tool for understanding electron localization in atoms.

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