Concept explainers
Mathematically, the uncertainty
Interpretation:
The value of
Concept introduction:
The uncertainty principle deals only with the parameters that cannot be measured simultaneously. Such two parameters are momentum and position in same direction. The uncertainty principle is given by formula as follows.
Where,
•
•
•
Answer to Problem 10.71E
The uncertainty in the position of a particle is
The uncertainty in the momentum of a particle is
The uncertainty principle has been validated.
Explanation of Solution
From Appendix
The average value of
Where,
•
•
The given wavefunction
The position operator is defined as
Substitute the value of
Assume
Substitute the value of
Therefore, the average value of position,
From Appendix
The average value of
Where,
•
•
The given wavefunction
The operator
Substitute the value of
Assume
Substitute the value of
Therefore, the value of
From Appendix
The average value of
Where,
•
•
The wavefunction at
The wavefunction at
The operator
Substitute the value of
Assume
Substitute the value of
Therefore, the value of
From Appendix
The average value of
Where,
•
•
The wavefunction at
The wavefunction at
The operator
Substitute the value of
Assume
Substitute the value of
Therefore, the value of
The given relation is represented as,
Where,
•
The equation (9) can be represented for the value of
Substitute the value of
Therefore, the uncertainty in the position of a particle is
The equation (9) can be represented for the value of
Substitute the value of
Therefore, the uncertainty in the momentum of a particle is
The uncertainty principle is given by formula as follows.
Where,
•
•
•
Substitute the value of
Substitute the value of
The value of left hand side in the above expression is greater than the value of right-hand side. Therefore, the above expression shows that the uncertainty principle is valid.
The uncertainty in the position of a particle is
The uncertainty in the momentum of a particle is
The uncertainty principle has been validated.
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Chapter 10 Solutions
Physical Chemistry
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