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Describe the ground state of hydrogen in terms of wave function, probability density, and atomic orbitals.
Distinguish between Bohr’s and Schrödinger’s model of the hydrogen atom. In particular, compare the energy and orbital angular momentum of the ground states.
The Bohr model describes the electron as a particle that moves around the proton in well-defined orbits. Schrödinger’s model describes the electron as a wave, and knowledge about the position of the electron is restricted to probability statements. The total energy of the electron in the ground state (and all excited states) is the same for both models. However, the orbital angular momentum of the ground state is different for these models. In Bohr’s model, , and in Schrödinger’s model, .
The wave function is evaluated at rectangular coordinates ( ) (2, 1, 1) in arbitrary units. What are the spherical coordinates of this position?
If an atom has an electron in the state with , what are the possible values of l ?
What are the possible values of m for an electron in the state?
are possible
What, if any, constraints does a value of place on the other quantum numbers for an electron in an atom?
What are the possible values of m for an electron in the state?
are possible
(a) How many angles can L make with the z -axis for an electron? (b) Calculate the value of the smallest angle.
The force on an electron is “negative the gradient of the potential energy function.” Use this knowledge and [link] to show that the force on the electron in a hydrogen atom is given by Coulomb’s force law.
What is the total number of states with orbital angular momentum ? (Ignore electron spin.)
The wave function is evaluated at spherical coordinates where the value of the radial coordinate is given in arbitrary units. What are the rectangular coordinates of this position?
(1, 1, 1)
Coulomb’s force law states that the force between two charged particles is:
Use this expression to determine the potential energy function.
Write an expression for the total number of states with orbital angular momentum l .
For the orbital angular momentum quantum number,
l , the allowed values of:
.
With the exception of
, the total number is just 2
l because the number of states on either side of
is just
l . Including
, the total number of orbital angular momentum states for the orbital angular momentum quantum number,
l , is:
Later, when we consider electron spin, the total number of angular momentum states will be found to twice this value because each orbital angular momentum states is associated with two states of electron spin: spin up and spin down).
Consider hydrogen in the ground state, . (a) Use the derivative to determine the radial position for which the probability density, P ( r ), is a maximum.
(b) Use the integral concept to determine the average radial position. (This is called the expectation value of the electron’s radial position.) Express your answers into terms of the Bohr radius, . Hint: The expectation value is the just average value. (c) Why are these values different?
What is the probability that the 1 s electron of a hydrogen atom is found outside the Bohr radius?
The probability that the 1 s electron of a hydrogen atom is found outside of the Bohr radius is
How many polar angles are possible for an electron in the state?
What is the maximum number of orbital angular momentum electron states in the shell of a hydrogen atom? (Ignore electron spin.)
For , (1 state), and (3 states). The total is 4.
What is the maximum number of orbital angular momentum electron states in the shell of a hydrogen atom? (Ignore electron spin.)
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