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mathfilip
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Im reading in a quantum mechanics book and need help to show the following relationship, (please show all the steps):
If A,B,C are operators:
[A,BC] = B[A,C] + [A,B]C
If A,B,C are operators:
[A,BC] = B[A,C] + [A,B]C
mathfilip said:Im reading in a quantum mechanics book and need help to show the following relationship, (please show all the steps):
If A,B,C are operators:
[A,BC] = B[A,C] + [A,B]C
stevenb said:Just expand out the commutation operators based on the definition, i.e. [X,Y]=XY-YX.
When you see how easy this is, you will laugh and might even be embarrassed that you posted such an easy question - no offense intended ... I've done much worse.
The commutation relation between two operators A and B is defined as the mathematical relationship between the operators when they act on a given state or system. It is denoted by [A,B] and is equal to the operator AB minus the operator BA. The result of this operation is a new operator that describes the relationship between A and B.
The commutation relation is related to the uncertainty principle in quantum mechanics. The uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. This is because the commutation relation between position and momentum operators is non-zero, meaning they do not commute. Therefore, they cannot have simultaneous well-defined values.
The commutation relation is significant in quantum mechanics as it helps us understand the fundamental properties of particles and systems. It allows us to determine whether two operators can be measured simultaneously and if not, how they are related. It also plays a crucial role in the formulation of Heisenberg's uncertainty principle and helps us understand the limitations of our ability to measure physical quantities in the quantum world.
The commutation relations follow certain rules under multiplication and addition. When two operators are multiplied, their commutator is also multiplied by a constant factor. For example, if [A,B] = C, then [2A,3B] = 6C. However, when two operators are added, their commutator is not affected. For example, if [A,B] = C, then [A+C,B] = [A,B].
Yes, the commutation relation between two operators can change over time. In quantum mechanics, operators can change over time through the process of time evolution. As a result, the commutation relation between two operators can also change, and this can have implications for the measurement of physical quantities and the behavior of systems over time.