Sriman Dutta Posted November 23, 2020 Share Posted November 23, 2020 On 11/21/2020 at 6:27 AM, joigus said: Conjugate variables are certainly peculiar. Their properties cannot be simulated by any finite-dimensional space of states and thereby cannot be completely understood with discrete mathematics. They are the domain of transcencental mathematics. Unlike the famous \( J_x \), \( J_y \), \( J_z \) that people use in all the completeness theorems, they always pair in couples, one of which is conserved, the other is not. Indeed. And the fascinating thing is how their operators' commutator give a constant ! Like take x and k, the k-observable will have a d/dx operator form in x-space and hence their commutator will yield a constant. Conversely, you have d/dk operator form for x-observable in k-space, and again their commutator is a constant. Tricky maths! On 11/21/2020 at 3:22 PM, studiot said: Yes QM is still very much a work in progress/unfinished business. +1 Pretty much. And I guess I read somewhere that this immediate state collapse violates the mximum speed postulate in relativity. I might be wrong, but I guess that this immediate effect of any field (like Newtonian gravity) had the drawback of effects coming instantly, yet any information can at most travel at c. Link to comment Share on other sites More sharing options...

joigus Posted November 23, 2020 Share Posted November 23, 2020 (edited) Ok, so here's the mathematical scoop. Copenhagen's interpretation of QM has 8 postulates (presentations may vary depending on the particular version). Two of them are concerned with how states change with time. One of them is called "evolution postulate" or "Schrödinger's equation"; the other is about so-called "measurements". Suppose there are only two states, \( \left|1\right\rangle \) and \( \left|2\right\rangle \). An arbitrary system in this simplified version of the world can be in an arbitrary superposition of both, \[\left(\begin{array}{c} c_{1}\\ c_{2} \end{array}\right)=c_{1}\left|1\right\rangle +c_{2}\left|2\right\rangle\] where \( c_1 \) and \( c_2 \) are complex numbers such that \( \left|c_{1}\right|^{2}+\left|c_{2}\right|^{2}=1 \). Whenever you write a state like this, a certain observable \( Q \) is implied, and \( c_1 \) and \( c_2 \) give you the probabilities that measuring this observable, the outcomes are either \( q_1 \) or \( q_2 \): \[\mathcal{P}\left(q_{1}\right)=\left|c_{1}\right|^{2}\] \[\mathcal{P}\left(q_{2}\right)=\left|c_{2}\right|^{2}\] A convenient way to express this observable \( Q \) is in term of its projectors, \[ Q=q_{1}P_{q_{1}}+q_{2}P_{q_{2}}=q_{1}\left(\begin{array}{cc} 1 & 0\\ 0 & 0 \end{array}\right)+q_{2}\left(\begin{array}{cc} 0 & 0\\ 0 & 1 \end{array}\right) \] But other projectors exist, because other observables exist. Projectors, in this simple case, are Hermitian --essentially real-- \( 2\times2 \) matrices satisfying, \[P^{2}=P\] For example, \[P_{r_{1}}=\left(\begin{array}{cc} \frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2} \end{array}\right)\] \[P_{r_{2}}=\left(\begin{array}{cc} \frac{1}{2} & -\frac{1}{2}\\ \frac{1}{2} & -\frac{1}{2} \end{array}\right)\] These ones correspond to a different observable, say \( R \) , incompatible with \( Q \) , \[R=r_{1}\left(\begin{array}{cc} \frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2} \end{array}\right)+r_{2}\left(\begin{array}{cc} \frac{1}{2} & -\frac{1}{2}\\ \frac{1}{2} & -\frac{1}{2} \end{array}\right)\] Observables can be seen as fundamental questions on the system, like 'what is the value of \( Q \)?' Projectors can be seen as observables themselves; answers to the questions 'is the value of \( Q \) equal to \( q_1 \)?' They are yes/no questions. What Copenhagen's QM measurement postulate tells you is that, when you \(Q\)-measure ('ask a \(Q\)-question') to the system, and the output happens to be, say, \( q_1 \), the salient state is, \[\left|\psi'\right\rangle =\frac{P_{q_{1}}\left|\psi\right\rangle }{\left\Vert P_{q_{1}}\left|\psi\right\rangle \right\Vert }=\frac{1}{\left|c_{1}\right|}\left(\begin{array}{c} c_{1}\\ 0 \end{array}\right)=e^{i\alpha}\left(\begin{array}{c} 1\\ 0 \end{array}\right)\] OTOH, what the Schrödinger equation tells you, in a nutshell, is that the most general form of evolution is given by, \[\left(\begin{array}{c} c_{1}'\\ c_{2}' \end{array}\right)=\left(\begin{array}{cc} u_{11} & u_{12}\\ u_{21} & u_{22} \end{array}\right)\left(\begin{array}{c} c_{1}\\ c_{2} \end{array}\right)\] where, \[u_{11}u_{22}-u_{12}u_{21}=e^{i\alpha}\] \( e^{i\alpha} \) is the way you parametrize a complex number of modulus \( 1 \). This is the complex-numbers equivalent to a rotation. This \( U \) matrix is not an observable; it's an evolution operator. And it doesn't have to be real; it must be 'unitary' –conserve probability–. Now, it is impossible to equate the change of state posited in the projection postulate to any \( U \). That is, in a nutshell, the problem of measurement. Very mathematically, if very simply --I hope--, stated. Edited November 23, 2020 by joigus Link to comment Share on other sites More sharing options...

studiot Posted November 23, 2020 Share Posted November 23, 2020 On 11/18/2020 at 1:17 AM, Anchovyforestbane said: I'm into QM, and science in general, because I have a passion for understanding and utilizing the complex systems and structures composing our universe. The Copenhagen Interpretation is the most popular way of understanding the probabilistic nature of nuclear physics, but I do not find the Many-Worlds Corollary to be a reasonable conclusion. Markus and Joigus have offered some mathematical insights to uncertainty but it is also useful to know that both the maths and the physics say the same thing. Just as both maths and physics says that distance = speed x time, Maths offers this as an algebraic fact, whilst Physics considers the (physical) meaning of the variables and equations concerned, including the dimensional analysis of those variables and mathematical statements. So in Spectroscopy we observe that the single frequency line spectra are not actually perfectly single frequency, they are slightly blurred. This can be interpreted as the time taken for a system to actually perform the transition that absorbs or emits radiation. 1 Link to comment Share on other sites More sharing options...

Orange6 Posted November 23, 2020 Share Posted November 23, 2020 (edited) On 11/17/2020 at 10:30 PM, Anchovyforestbane said: Is that so? Any book I've read matches them together as though one must necessarily imply the other. Copenhagen and MWI are competing interpretations. Copenhagen and MWI both have problems that seemingly aren't very well resolved, as do the more minor QM interpretations. Copenhagen has the problem of defining exactly what an observation is. MWI has problems of probability. (Everything occurs somewhere so how can you assign probability.) All interpretations of quantum mechanics, frankly, are pretty bad. Really, the question is about which interpretation of quantum mechanics is least bad. It currently is untestable which interpretation is correct, and perhaps it always will be untestable. Edited November 23, 2020 by Orange6 Link to comment Share on other sites More sharing options...

studiot Posted November 23, 2020 Share Posted November 23, 2020 17 minutes ago, Orange6 said: Copenhagen and MWI are competing interpretations. Copenhagen and MWI both have problems that seemingly aren't very well resolved, as do the more minor QM interpretations. Wave and corpuscular hypotheses are competing interpretations of light. That doesn't make them mutually exclusive, that a significant point of QM. So can you demonstrate that Cop and MW are mutually exclusive ? Otherwise why does it matter that they' compete' ? 20 minutes ago, Orange6 said: Copenhagen has the problem of defining exactly what an observation is. MWI has problems of probability. (Everything occurs somewhere so how can you assign probability.) Seems you understand neither of these imperfect interpretations enough to make these comments. Link to comment Share on other sites More sharing options...

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