Calculate the probability that the electron in the hydrogen atom, in its ground state, will be found between spherical shells whose radii are a and 2a , where a is the Bohr radius?
The required probability is P=0.439.
The given data is listed below.
The ground state wave function of hydrogen atom is given by,
Here, the Bohr radius is .
The probability of finding the electron found between spherical shells is,
Hence, the probability of the electron in the hydrogen atom in its ground state is 0.439.
The radial probability density for the ground state of the hydrogen atom is a maximum when r = a , where is the Bohr radius. Show that the average value of r, defined as
has the value 1.5a. In this expression for , each value of (P)r is weighted with the value of r at which it occurs. Note that the average value of is greater than the value of r for which (P)r is a maximum.
The wave functions for the three states with the dot plots shown in Fig. 39-23, which have n = 2 , l = 1 , and 0, and , are
in which the subscripts on give the values of the quantum numbers n , l , and the angles and are defined in Fig. 39-22. Note that the first wave function is real but the others, which involve the imaginary number i, are complex. Find the radial probability density P(r) for (a) and (b) (same as for ). (c) Show that each P(r) is consistent with the corresponding dot plot in Fig. 39-23. (d) Add the radial probability densities for , , and and then show that the sum is spherically symmetric, depending only on r .
A hydrogen atom is excited from its ground state to the state with n=4. (a) How much energy must be absorbed by the atom? Consider the photon energies that can be emitted by the atom as it de-excites to the ground state in the several possible ways. (b) How many different energies are possible; What are the (c) highest, (d) second highest, (e) third highest, (f) lowest, (g) second lowest, and (h) third lowest energies.
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