Associativity of operators in quantum mechanics

In summary, the correct interpretation of < \frac{\partial {A}}{\partial t} > is a set of operators dependent on a parameter, t, and its derivative is defined through a limiting procedure in the presence of vectors in the domain of all A(t). The expectation value is given by < \phi, \frac{dA(t)}{dt}\phi>.
  • #1
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Homework Statement


What is the correct interpretation of
[tex] < \frac{\partial {A}}{\partial t} >[/tex], where A is an operator?

Homework Equations


for a wave function [tex]\phi[/tex] and operator A,
[tex]<A> = \int_{V}\phi^{*}(A\phi)dV[/tex]

The Attempt at a Solution


I thought it could mean
[tex] < \frac{\partial {A}}{\partial t} > = \int_{V}\phi^{*}\frac{\partial}{\partial t}(A\phi)dV[/tex]
but then again it might mean
[tex] < \frac{\partial {A}}{\partial t} > = \int_{V}\phi^{*}(\frac{\partial A}{\partial t})(\phi)dV[/tex].

I read an article saying that operators are associative. But, when I think about the operators [tex]t[/tex] and [tex]\frac{\partial}{\partial t}[/tex], then,

[tex]\frac{\partial}{\partial t}\left(t\phi\right) = t\frac{\partial\phi}{\partial t} + \phi \neq \left(\frac{\partial t}{\partial t}\right)\phi = \phi[/tex]

any thoughts?
 
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  • #2
The d/dt is not an operator in the normal sense. A(t) is interpreted as a set of operators which depend on a parameter, t. The derivative of this A(t) wrt the parameter is defined by means of a limiting procedure always in the presence of vectors in the domain of all A(t).

[tex] \frac{d A(t)}{dt} \psi = \lim_{t\rightarrow 0} \frac{A(t)\psi - A(0)\psi}{t} [/tex]

The expectation value is then simply [itex] \left\langle \phi, \frac{dA(t)}{dt}\phi\right\rangle [/itex]
 

1. How does associativity of operators affect calculations in quantum mechanics?

The associativity of operators in quantum mechanics refers to the order in which operators are applied in a calculation. This is important because the result of a calculation can vary depending on the order of operations. In quantum mechanics, operators represent physical quantities such as position, momentum, and energy. Therefore, getting the correct order of operations is crucial in obtaining accurate results.

2. What are some examples of operators that exhibit associativity in quantum mechanics?

Some common examples of operators that exhibit associativity in quantum mechanics include the position operator, momentum operator, and energy operator. These operators follow the rules of standard algebra, where the order of multiplication does not affect the result. However, in quantum mechanics, the order in which operators are applied can significantly impact the outcome of a calculation.

3. How is the associativity of operators related to the uncertainty principle in quantum mechanics?

The uncertainty principle in quantum mechanics states that it is impossible to know the exact values of certain pairs of physical quantities, such as position and momentum, simultaneously. This is due to the non-commutativity of these operators, which means that the order in which they are applied affects the result. The associativity of operators plays a crucial role in understanding and calculating the uncertainty principle in quantum mechanics.

4. Can the associativity of operators be violated in quantum mechanics?

No, the associativity of operators cannot be violated in quantum mechanics. This is because operators represent physical quantities that follow the laws of standard algebra. Therefore, the order of operations must be taken into account to obtain accurate results. Violating the associativity of operators would lead to incorrect calculations and predictions, which goes against the fundamental principles of quantum mechanics.

5. How does the associativity of operators impact the interpretation of experimental results in quantum mechanics?

The associativity of operators is crucial in interpreting experimental results in quantum mechanics. The order in which operators are applied in a calculation can affect the outcome, leading to different interpretations of the data. Additionally, the non-commutativity of certain operators also plays a role in the interpretation of experimental results, as it can influence the uncertainty and accuracy of measurements. Understanding and correctly applying the associativity of operators is essential in accurately interpreting experimental data in quantum mechanics.

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