If the commutation between the two operators Oˆ1 and Oˆ2 vanishes, then show that their inverses also commute.
Understand the Problem
The question is asking to demonstrate a mathematical property regarding operators. Specifically, it states that if the commutation of two operators Oˆ1 and Oˆ2 is zero (meaning they commute), we need to show that their inverses also commute. This requires understanding the properties of operator inverses and the implications of commutation in a mathematical framework.
Answer
If $O_1$ and $O_2$ commute, then $O_1^{-1}$ and $O_2^{-1}$ also commute: $O_1^{-1} O_2^{-1} = O_2^{-1} O_1^{-1}$.
Answer for screen readers
If $O_1$ and $O_2$ commute, then their inverses $O_1^{-1}$ and $O_2^{-1}$ also commute, meaning: $$ O_1^{-1} O_2^{-1} = O_2^{-1} O_1^{-1} $$
Steps to Solve
- Define Commuting Operators
Two operators $O_1$ and $O_2$ are said to commute if their commutation relation is given by: $$ [O_1, O_2] = O_1 O_2 - O_2 O_1 = 0 $$
- Assume the Commutation Relation
Given the problem states that the commutation of the operators is zero: $$ O_1 O_2 = O_2 O_1 $$
- Multiply by Inverses
To show that their inverses commute, let's denote the inverses of the operators as $O_1^{-1}$ and $O_2^{-1}$. We need to check if: $$ O_1^{-1} O_2^{-1} = O_2^{-1} O_1^{-1} $$
- Manipulate the Expression
We can rewrite the expression using the identity operator $I = O_1 O_1^{-1}$ and $I = O_2 O_2^{-1}$. Multiply both sides by $O_1 O_2$: $$ O_1 O_1^{-1} O_2 O_2^{-1} = O_1 O_2 O_2^{-1} O_1^{-1} $$
- Apply Associative Property
By applying the associative property: $$ O_2 = O_1 O_2 O_1^{-1} O_1 $$ becomes: $$ O_2 = O_2 O_1 O_1^{-1} $$
- Reach to Final Conclusion
Since $O_1 O_2 = O_2 O_1$, it implies: $$ O_1^{-1} O_2 O_1^{-1} = O_2^{-1} O_1 $$
Thus, we establish: $$ O_1^{-1} O_2^{-1} = O_2^{-1} O_1^{-1} $$
Therefore, the inverses also commute.
If $O_1$ and $O_2$ commute, then their inverses $O_1^{-1}$ and $O_2^{-1}$ also commute, meaning: $$ O_1^{-1} O_2^{-1} = O_2^{-1} O_1^{-1} $$
More Information
The result demonstrated shows a vital property of operators in quantum mechanics, highlighting the relationship between the commutation of operators and their inverses. This property is essential in various applications, particularly in quantum mechanics, where the concepts of observable quantities and their measurable states play a critical role.
Tips
- Confusing the properties of operators: It is essential to correctly apply definitions and properties of inverses and commutation.
- Incorrect assumption about associativity without justification: Remember that multiplication is associative for operators in this framework.
