The uncertainty in measuring the radial position of electron staying in the ground state of Hydrogen.
Answer to Problem 77AP
The uncertainty in measuring the radial position of electron of Hydrogen is
Explanation of Solution
Write the expression for the average of the square of distance between electron and proton.
Here,
Use expression (I) to find the average of square of distance between electron and proton of the Hydrogen atom in
Here,
Write the expression for the wave function of Hydrogen atom in
Here,
Write the expression for complex conjugate of
Write the expression for the volume of the spherical
Here,
Take the derivative of volume with respect to the radius of the sphere and obtain
Use expression (III), (IV) and (VI) in (II).
Simplify equation (VII) and evaluate the integral from zero to infinity.
Integral in equation (VIII) is similar to the general form
Compare expressions (VIII) and (IX).
Use expressions (VIII) and (IX) to find the solution of (VIII).
Write the expression for the root mean square uncertainty in
Write the expression to find
Use expression (III), (IV), (VI) in expression (XIII) to find
Simplify equation (XIII) and integrate from zero to infinity.
Compare expressions (XIV) and (IX).
Use expression (IX) to solve (XIII).
Conclusion:
Substitute
Therefore, the uncertainty in measuring the radial position of electron of Hydrogen is
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Chapter 42 Solutions
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