In the beginning of the twentieth century, some scientists thought that a nucleus may contain both electrons and protons. Use the Heisenberg uncertainty principle to show that an electron cannot ha confined within a nucleus. Repeat the calculation for a proton Comment on your results Assume the radius of a nucleus. to be
Interpretation:
The statement, “An electron cannot be confined within the nucleus”, and the statement, “A proton cannot be confined within the nucleus”, are to be proved by using the Heisenberguncertaintyprinciple.
Concept Introduction:
According to the Heisenberg uncertainty principle, the product of the uncertainty in position and momentum of a particle cannot be less than
Here,
As momentum is the product of mass and velocity, the equation of Heisenberg uncertainty principle can also be written as follows:
Here,
Answer to Problem 52QP
Solution: The calculated speed of anelectron is higher than the speed of light, so it cannot be confined within the nucleus. The uncertainty in velocity of the electron is
The value of uncertainty of a proton is less than the speed of light, so a proton cannot be confined within the nucleus. The uncertainty in velocity of the proton is
Explanation of Solution
Given information:
The mass of an electron
The mass of a proton
The radius of the nucleus is the uncertainty in position that is
The uncertainty in velocity of an electron can be evaluated as follows:
Substitute
Therefore, the uncertainty invelocity of the electron is
This value for the uncertainty is impossible as it exceeds the speed of light. If the electron exists in the nucleus, then it must travel with a speed of
The uncertainty in velocity of the proton can be evaluated as follows:
Substitute
Therefore, the uncertainty invelocity of the proton is
The value of uncertainty of proton is less than the speed of light, so a proton cannot be confined within the nucleus.
When the electron exists in the nucleus, it must travel with a speed of
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