tag:blogger.com,1999:blog-3438397030323497045.post3323640051693838801..comments2014-07-18T04:06:54.195-07:00Comments on The (Epistemological) Foundations of Physics: About the meaning of Schrödinger EquationAnssi Hyytiäinennoreply@blogger.comBlogger3125tag:blogger.com,1999:blog-3438397030323497045.post-13745104786279726222014-07-18T04:06:54.195-07:002014-07-18T04:06:54.195-07:00Regarding MrQuincle's comment, “Would you also...Regarding MrQuincle's comment, “Would you also like to explain the step from 3.8 to 3.9.” together with “The function g(x) seems quite complicated and is related to V(x) in the end.” it may be that he is confusing dVr-1 with V(x), two very different references. The expression dVr-1 refers to the differential volume associated with the x variables in the “remaining” collection; see equation (3.5). The “minus 1” refers to the fact that x sub one is not being integrated over!<br /><br />The V(x) defined in equation (3.15) is an entirely different thing. That expression is no more than the consequence of identifying g(x) (the result of the integration) with Schrodinger's V(x). It essentially inserts the consequence of that integration over the rest of the universe with what is ordinarily seen as potential energy.<br /><br />I hope that clears things up.Richard Staffordhttp://www.blogger.com/profile/04889640688995133348noreply@blogger.comtag:blogger.com,1999:blog-3438397030323497045.post-64518524510074015432014-07-17T12:32:54.056-07:002014-07-17T12:32:54.056-07:00Thank you for the comment.
There is really no &qu...Thank you for the comment.<br /><br />There is really no "normal" derivation of Schrödinger equation, as historically it is directly based on some postulates of quantum mechanics. That is the meaning of the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#The_wave_equation_for_particles" rel="nofollow">Feynman quote on Wikipedia:</a><br /><br />"Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger."<br /><br />When Schrödinger created the equation, he was simply motivated by <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/debrog.html#c3" rel="nofollow">de Broglie's notions</a> and other associated developments in physics at that time. Motivated by these ideas, and some guess work, he came up with an equation that replicates observational data, but he did not really know why.<br /><br />If you google for a derivation of Schrödinger equation, you will get various presentations deconstructing some algebraic connections to Newtonian definitions. For instance I spotted <a href="http://journeymanphilosopher.blogspot.com/2011/05/trying-to-understand-schrodingers.html" rel="nofollow">this blog post</a> which is related to that issue. See also the video link at Addendum 5 in that page.<br /><br />At any rate, within the current understanding of physics it would be said that Schrödinger equation cannot be reduced to anything more fundamental. That is no longer exactly true, as it appears to arise from constraints that govern sensible object definitions.<br /><br />This just entails a complete paradigm shift from the idea that we are trying to explain a world made of persistent particles that behave like waves when we are not looking, to the idea that defined particles are in fact part of a mental terminology useful for representing some information (reality) in meaningful manner.<br /><br />$V(x)$ in Schrödinger equation represents potential energy in location x. It can be used to model some external influence, for instance to<br /><a href="http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html#c1" rel="nofollow">model some boundaries for the particle</a>.<br /><br />In the analysis, $g(\vec{x})$ is a shorthand to an integral that represents the impact of the rest of the universe, given the location of the single element of interest. It is something which can be expressed as a function of single $x$ (that single element of interest), if a solution to the rest of the universe is known or the rest of the universe can be ignored. To understand that bit better, see how the function $F$ arose in (3.4), and also see the additional clarifying comments related to that function on page 52.<br /><br />If people want, this could be perhaps explained in greater detail in a separate post.<br /><br />But so effectively $g$ is embedded inside the $V$ and they are indeed associated to the same idea (representing external influences).<br /><br />You should also read from the book the few pages following directly after the Schrödinger equation (3.15), which further clarifies some details surrounding this equation, and is somewhat relevant to your questions regarding its "normal" derivation.Anssi Hyytiäinenhttp://www.blogger.com/profile/16836638486221364670noreply@blogger.comtag:blogger.com,1999:blog-3438397030323497045.post-24782919136633660882014-07-16T14:49:18.546-07:002014-07-16T14:49:18.546-07:00Hi Anssi,
Perhaps the "normal" derivat...Hi Anssi, <br /><br />Perhaps the "normal" derivation of the Schrodinger equation would be great to have in parallel.<br /><br />Would you also like to explain the step from 3.8 to 3.9. The function g(x) seems quite complicated and is related to V(x) in the end. Does V(x) not normally have some physical meaning (other than the assumptions of the rest of the universe not influencing the system, the rest of the universe is stationary in time, and the time derivative of the wave function is equal to the wave function times an imaginary constant times q)?MrQuinclehttp://www.blogger.com/profile/12622264993546058205noreply@blogger.com