- #1
Vyurok
- 7
- 1
In the book Lectures on Quantum Mechanics by L.D. Faddeev and O.A. Yakubovskii the authors write the following:
Among the observables there may be some that are functionally dependent, and hence it is necessary to impose a condition on the probability distributions of such observables. If an observable ##\varphi## is a function of an observable ##f##, ##\varphi = \varphi(f)##, then this assertion means that a measurement of the numerical value of ##f## yielding a value ##f_0## is at the same time a measurement of the observable ##\varphi## and gives for it the numerical value ##\varphi_0 = \varphi(f_0)##. Therefore, ##\omega_f(E)## and ##\omega_{\varphi(f)}(E)## are connected by the equality
$$\omega_{\varphi(f)}(E) = \omega_f\big(\varphi^{-1}(E)\big),$$
I have a question:
What if ##\phi = \phi(f_{1}, ..., f_{n})##? How then is ##\omega_{\phi}## expressed in terms of ##\omega_{f_{i}}## where ##(i = 1, …, n)##?
Further:
For the mean values of observables we require the following conditions, which are natural from a physical point of view:
$$
\begin{aligned}
&1) && \langle C \mid \omega \rangle = C, \\
&2) && \langle f + \lambda g \mid \omega \rangle = \langle f \mid \omega \rangle + \lambda \langle g \mid \omega \rangle, \\
&3) && \langle f^2 \mid \omega \rangle \geq 0.
\end{aligned}
$$
And this I have a question:
Why is it stated here that these conditions must be required? The first and third hold automatically by the definition of a state, and the second might also hold, provided we can correctly define how ##\omega_{\phi(f_{1}, ..., f_{n})}## should be expressed in terms of ##\omega_{f_{i}}## where ##(i = 1, …, n)##. What, then, was meant here?
And further:
If these requirements are introduced, then the realization of the algebra of observables itself determines a way of describing the states
What do these words mean?
And here's another thing:
Indeed, the mean value is a positive linear functional on the algebra ##\mathfrak{A}## of observables. The general form of such a functional is
$$\langle f \mid \omega \rangle = \int_{\mathcal{M}} f(p, q) \, d\mu_\omega(p, q),$$
where ##d\mu_\omega(p, q)## is the differential of the measure on the phase space, and the integral is over the whole of phase space. It follows from the condition 1) that
$$\int_{\mathcal{M}} d\mu_\omega(p, q) = \mu_\omega(\mathcal{M}) = 1.$$
It's not clear to me:
What do ##p## and ##q## mean here? They definitely cannot be position and momentum, because in quantum mechanics it is impossible to measure them simultaneously with absolute precision. What is meant here by phase space? In what exact way is a measure ##\mu_{\omega}## on phase space assigned to a state? How is an observable ##f## associated with a function ##f(q, p)## on the phase space? Where does formula (7) come from, and what does it mean? Why is (8) valid?
Among the observables there may be some that are functionally dependent, and hence it is necessary to impose a condition on the probability distributions of such observables. If an observable ##\varphi## is a function of an observable ##f##, ##\varphi = \varphi(f)##, then this assertion means that a measurement of the numerical value of ##f## yielding a value ##f_0## is at the same time a measurement of the observable ##\varphi## and gives for it the numerical value ##\varphi_0 = \varphi(f_0)##. Therefore, ##\omega_f(E)## and ##\omega_{\varphi(f)}(E)## are connected by the equality
$$\omega_{\varphi(f)}(E) = \omega_f\big(\varphi^{-1}(E)\big),$$
I have a question:
What if ##\phi = \phi(f_{1}, ..., f_{n})##? How then is ##\omega_{\phi}## expressed in terms of ##\omega_{f_{i}}## where ##(i = 1, …, n)##?
Further:
For the mean values of observables we require the following conditions, which are natural from a physical point of view:
$$
\begin{aligned}
&1) && \langle C \mid \omega \rangle = C, \\
&2) && \langle f + \lambda g \mid \omega \rangle = \langle f \mid \omega \rangle + \lambda \langle g \mid \omega \rangle, \\
&3) && \langle f^2 \mid \omega \rangle \geq 0.
\end{aligned}
$$
And this I have a question:
Why is it stated here that these conditions must be required? The first and third hold automatically by the definition of a state, and the second might also hold, provided we can correctly define how ##\omega_{\phi(f_{1}, ..., f_{n})}## should be expressed in terms of ##\omega_{f_{i}}## where ##(i = 1, …, n)##. What, then, was meant here?
And further:
If these requirements are introduced, then the realization of the algebra of observables itself determines a way of describing the states
What do these words mean?
And here's another thing:
Indeed, the mean value is a positive linear functional on the algebra ##\mathfrak{A}## of observables. The general form of such a functional is
$$\langle f \mid \omega \rangle = \int_{\mathcal{M}} f(p, q) \, d\mu_\omega(p, q),$$
where ##d\mu_\omega(p, q)## is the differential of the measure on the phase space, and the integral is over the whole of phase space. It follows from the condition 1) that
$$\int_{\mathcal{M}} d\mu_\omega(p, q) = \mu_\omega(\mathcal{M}) = 1.$$
It's not clear to me:
What do ##p## and ##q## mean here? They definitely cannot be position and momentum, because in quantum mechanics it is impossible to measure them simultaneously with absolute precision. What is meant here by phase space? In what exact way is a measure ##\mu_{\omega}## on phase space assigned to a state? How is an observable ##f## associated with a function ##f(q, p)## on the phase space? Where does formula (7) come from, and what does it mean? Why is (8) valid?