22. Define the operator B = câ + p and Â= câ - p. Calculate Â,B.
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![**Problem Statement:**
22. Define the operator \( \hat{B} = c\hat{x} + \hat{p} \) and \( \hat{A} = c\hat{x} - \hat{p} \). Calculate \( [\hat{A}, \hat{B}] \).
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**Discussion:**
The task is to calculate the commutator \([\hat{A}, \hat{B}]\) where:
- \( \hat{B} = c\hat{x} + \hat{p} \)
- \( \hat{A} = c\hat{x} - \hat{p} \)
In quantum mechanics, the commutator \([ \hat{A}, \hat{B} ]\) is defined as:
\[
[\hat{A}, \hat{B}] = \hat{A} \hat{B} - \hat{B} \hat{A}
\]
With the provided definitions, you can proceed to explicitly calculate the commutator using the properties of operators \(\hat{x}\) (position operator) and \(\hat{p}\) (momentum operator), and their known commutation relation \( [\hat{x}, \hat{p}] = i\hbar \). This can include expansions and simplifications by collecting like terms, considering these operators' linearity and distributive properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e50119e-8646-4255-90fd-98958ba58941%2Fea15754e-86f8-443d-9389-7cd271def397%2Flchkg7p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
22. Define the operator \( \hat{B} = c\hat{x} + \hat{p} \) and \( \hat{A} = c\hat{x} - \hat{p} \). Calculate \( [\hat{A}, \hat{B}] \).
---
**Discussion:**
The task is to calculate the commutator \([\hat{A}, \hat{B}]\) where:
- \( \hat{B} = c\hat{x} + \hat{p} \)
- \( \hat{A} = c\hat{x} - \hat{p} \)
In quantum mechanics, the commutator \([ \hat{A}, \hat{B} ]\) is defined as:
\[
[\hat{A}, \hat{B}] = \hat{A} \hat{B} - \hat{B} \hat{A}
\]
With the provided definitions, you can proceed to explicitly calculate the commutator using the properties of operators \(\hat{x}\) (position operator) and \(\hat{p}\) (momentum operator), and their known commutation relation \( [\hat{x}, \hat{p}] = i\hbar \). This can include expansions and simplifications by collecting like terms, considering these operators' linearity and distributive properties.
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