The wave function for the hydrogen-atom quantum state represented by the dot plot shown in Fig. 39-21, which has n = 2 and , is
in which a is the Bohr radius and the subscript on gives the values of the quantum numbers . (a) Plot and show that your plot is consistent with the dot plot of Fig. 39-21. (b) Show analytically that has a maximum at . (c) Find the radial probability density for this state. (d) Show that
and thus that the expression above for the wave function has been properly normalized.
b. It is proved that has a maximum at r = 4a.
c. The radial probability is .
d. It is proved that .
The wave function is given by,
Take the values of r on the horizontal axis, but actually, the values of r/a must be taken on the horizontal axis. Here, a is the Bohr radius, and it is a constant value.
The plot is shown in the below figure.
From the above figure, it can be observed that there is a high central peak between r = 0 and r = 2A. At r = 2a, the value of the wave function must be equal to zero . Also, the low peak value is reached its maximum value at r = 4a. The graph is not consistent with the dot plot of the given figure.
Differentiate the given wave function and equate to zero to find the maximum.
Therefore, it is proved that has a maximum at r = 4a.
The expression for the radial probability is as follows.
Therefore, the radial probability for the given state is .
Show that the value of as follows.
Therefore, it is proved that .
Consider a conduction electron in a cubical crystal of a conducting material. Such an electron is free to move throughout the volume of the crystal but cannot escape to the outside. It is trapped in a three-dimensional infinite well. The electron can move in three dimensions so that its total energy is given by
in which are positive integer values. Calculate the energies of the lowest five distinct states for a conduction electron moving in a cubical crystal of edge length .
An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energy difference equal to the energy of the n = 5 level? (c) Show that no pair of adjacent levels has an energy difference equal to the energy of the n = 6 level.
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