For the cubic 3D infinite well wave function
show that the correct normalization constant is .
The normalization constant is .
Cubic 3D infinite well wave function is given as:
The normalization condition can be expressed as:
We know that the probability of finding the particle will always be 1.
Evaluate the normalization condition for a given 3D infinite well solution to determine the value of the normalization constant A as:
For a 3D infinite well, its wave function is given by
L is the edge length of the square well.
We know that the probability density integrated over the volume of the 3D box must equal 1. Thus, we have
All three integrals multiplying each other are of the same form, for different values of so next evaluation just one of them for an arbitrary n.
Visualizing the dependence of the trigonometric functions graphically leads to the conclusion that,
Therefore, one segment can be solved as:
Thus, all three integrals multiplying each other in equation (1) are equal to each other and to .
Therefore, the equation (1) becomes,
Therefore, the normalization constant is .
Consider a 2D infinite well whose sides are of unequal length.
(a) Sketch the probability density as density of shading for the ground state.
(b) There are two likely choices for the next lowest energy. Sketch the probability density and explain how you know that this must be the next lowest energy. (Focus on the qualitative idea, avoiding unnecessary reference to calculations.)
An electron in a hydrogen atom is in the (n,l,ml) = (2,1,0) state.
(a) Calculate the probability that it would be found within 60 degrees of z-axis, irrespective of radius.
(b) Calculate the probability that it would be found between r = 2a0 and r = 6a0, irrespective of angle.
(c) What is the probability that it would be found within 60 degrees of the z-axis and between r = 2a0 and r = 6a0?
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