Wikipedia:Reference desk/Archives/Mathematics/2021 May 4

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May 4

Math for probability density function

 Resolved

Known facts:

$\tau (t)=e^{-{\frac {iEt}{\hbar }}}$ (1)

$\Psi (x,t)=\psi (x)\tau (t)=\psi (x)e^{-{\frac {iEt}{\hbar }}}$ (2)

$\exp \left({\overline {z}}\right)={\overline {\exp(z)}}$ (3)

$|\Psi (x,t)|^{2}=\Psi ^{*}(x,t)\Psi (x,t)$ (4)

${|\psi (x)|}^{2}=\psi ^{*}(x)\psi (x)$ (5)

I would like to confirm whether the following math is correct or not:

${\begin{aligned}|\Psi (x,t)|^{2}&=\Psi ^{*}(x,t)\Psi (x,t)\\&={\left[\psi (x)e^{-{\frac {iEt}{\hbar }}}\right]}^{*}\left[\psi (x)e^{-{\frac {iEt}{\hbar }}}\right]\\&={\psi }^{*}(x){\left(e^{-{\frac {iEt}{\hbar }}}\right)}^{*}\left[\psi (x)e^{-{\frac {iEt}{\hbar }}}\right]\\&={\psi }^{*}(x)e^{\frac {iEt}{\hbar }}\left[\psi (x)e^{-{\frac {iEt}{\hbar }}}\right]\\&={\psi }^{*}(x)e^{\frac {iEt}{\hbar }}\psi (x)e^{-{\frac {iEt}{\hbar }}}\\&={\psi }^{*}(x)\psi (x)\\&={|\psi (x)|}^{2}\end{aligned}}$

-- Justin545 (talk) 16:07, 4 May 2021 (UTC)

Is this in relation to the Born rule and the Schrödinger equation? There's plenty of quantum physics texts out there where you can check your work against. --Jayron32 16:13, 4 May 2021 (UTC)

I think you are right. But is there any free and online textbooks for quantum physics out there? - Justin545 (talk) 16:24, 4 May 2021 (UTC)

It might help if we knew your context. Where did you get this problem? --Jayron32 16:32, 4 May 2021 (UTC)

The PDF in this video [1] is for time-independent wavefunctions but the PDF in this section [2] is for time-dependent wavefunctions. -- Justin545 (talk) 16:44, 4 May 2021 (UTC)

And I got (1) from here [3]. -- Justin545 (talk) 16:58, 4 May 2021 (UTC)

Each step is valid. The first by (4); the second by the definition of $\Psi$; the third because complex conjugation distributes over multiplication; the fourth by the definition of complex conjugation; the fifth by the associativity of multiplication followed by (3), using that $\exp z$ and $e^{z}$ mean the same; the sixth is actually five sub-steps combined, the first sub-step using the commutativity of multiplication, the second an algebraic property of exponentiation (product of powers is power of sum), the third being that the sum of two opposite quantities simplifies to $0$, the fourth the property that $a^{0}=1,$ and the fifth sub-step that $1$ is the identity element of multiplication; the last by (5).  --Lambiam 20:35, 4 May 2021 (UTC)

That is a very detailed verification. Thank you for your help once again!

 Resolved -- Justin545 (talk) 19:19, 13 April 2021 (UTC)

I'd like to add that (2) is only valid if $\psi$ is an eigenfunction of the Hamilton operator, i.e. if the system is in an energy eigenstate. If this is not the case, i.e. if $\psi$ is a superposition or sum of energy eigenfunctions, then each has its own factor $\exp(-iE_{j}t/\hbar )$ and you get interference that makes $\Psi (x,t)$ time-dependent. --Wrongfilter (talk) 16:05, 5 May 2021 (UTC)

I assume you meant to say that then $|\Psi (x,t)|$ becomes time-dependent (which, obviously, implies that the identity $|\Psi (x,t)|=|\psi (x)|$ cannot hold).  --Lambiam 17:12, 5 May 2021 (UTC)

Yes, you're right. The time dependence in the overall phase factor has no observable consequences, that's why I tend to forget about it. But the absolute value is of course correct. --Wrongfilter (talk) 20:26, 5 May 2021 (UTC)

Ah, thanks for pointing it out. I apologize to you guys for not noticing that it's about the eigenequation

$\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+v(x)\right]\psi (x)=E\psi (x)$ (eq1)

from [4]. So I think modifications may be:

$\tau _{n}(t)=e^{-{\frac {iE_{n}t}{\hbar }}}$ (1a)

$\Psi _{n}(x,t)=\psi _{n}(x)\tau _{n}(t)=\psi _{n}(x)e^{-{\frac {iE_{n}t}{\hbar }}}$ (2a)

$|\Psi _{n}(x,t)|^{2}=\Psi _{n}^{*}(x,t)\Psi _{n}(x,t)$ (4a)

${|\psi _{n}(x)|}^{2}=\psi _{n}^{*}(x)\psi _{n}(x)$ (5a)

and therefore

${\begin{aligned}|\Psi _{n}(x,t)|^{2}&=\Psi _{n}^{*}(x,t)\Psi _{n}(x,t)\\&={\left[\psi _{n}(x)e^{-{\frac {iE_{n}t}{\hbar }}}\right]}^{*}\left[\psi _{n}(x)e^{-{\frac {iE_{n}t}{\hbar }}}\right]\\&={\psi }_{n}^{*}(x){\left(e^{-{\frac {iE_{n}t}{\hbar }}}\right)}^{*}\left[\psi _{n}(x)e^{-{\frac {iE_{n}t}{\hbar }}}\right]\\&={\psi }_{n}^{*}(x)e^{\frac {iE_{n}t}{\hbar }}\left[\psi _{n}(x)e^{-{\frac {iE_{n}t}{\hbar }}}\right]\\&={\psi }_{n}^{*}(x)e^{\frac {iE_{n}t}{\hbar }}\psi _{n}(x)e^{-{\frac {iE_{n}t}{\hbar }}}\\&={\psi }_{n}^{*}(x)\psi _{n}(x)\\&={|\psi _{n}(x)|}^{2}\end{aligned}}$

where each $\psi _{n}(x)$ is an eigenvector(eigenfunction) of (eq1) and $E_{n}$ is the eigenvalue(eigenenergy) corresponding to that eigenvector. -- Justin545 (talk) 19:48, 5 May 2021 (UTC)