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While deriving the Time Independent Schrodinger Equation, my book mentioned this line.

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So time and position of a particle are two independent variables. If they are equal to one another for all values of $t$ & $r$, then why should they be equal to a constant?

Can't we have other solutions to this other than treating both the sides as a constant?

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  • $\begingroup$ Have you taken a course on partial differential equations yet? This is explained in courses such as those $\endgroup$
    – Triatticus
    Sep 3, 2019 at 17:44
  • $\begingroup$ @Triatticus Actually no. My college is teaching me this stuff in the 1st semester itself. I have basic high school knowledge on Ordinary Differential Equations only. $\endgroup$ Sep 3, 2019 at 17:46
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    $\begingroup$ Can you think of another way for two functions of independent variables to be equal for all values of those variables? $\endgroup$
    – G. Smith
    Sep 3, 2019 at 17:47
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    $\begingroup$ I would suggest learning about separation of variables, this should help. $\endgroup$
    – user207248
    Sep 3, 2019 at 17:47
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    $\begingroup$ @harshit54: No, not really. FTs do pop up sometimes in PDE seps., to use the initial condition. But I'm not very good with FTs and had no problem following the method. It's kind of high school level. $\endgroup$
    – Gert
    Sep 3, 2019 at 18:25

2 Answers 2

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There are two logical options when you vary $t$: either the value of the left-hand side changes, or it doesn't. If it changes, then the right side must change as well, since they are equal. But the right-hand side can't change when you vary $t$, since it is not a function of $t$! Therefore, since varying $t$ produces no change in the left-hand-side, then the left-hand side must be constant. And since it is equal to the right-hand side, then they are both (the same) constant.

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    $\begingroup$ Well said. +1 from me. $\endgroup$
    – Gert
    Sep 3, 2019 at 18:00
  • $\begingroup$ Thanks for the answer. $\endgroup$ Sep 3, 2019 at 18:02
  • $\begingroup$ Also for this to be true, you assume that r and t are independent variables, right? $\endgroup$ Sep 7, 2019 at 14:14
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Can't we have other solutions to this other than treating both the sides as a constant?

No. The TDSE (and many other similar PDEs, like the TISE, the Newtonian wave equation and Fourier's equation) has a numeric separation constant. They are known as eigenvalue problems.

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