**Pre-script** (dated 26 June 2020): Our ideas have evolved into a full-blown realistic (or *classical*) interpretation of all things quantum-mechanical. In addition, I note the dark force has amused himself by removing some material. So no use to read this. Read my recent papers instead. đ

**Original post**:

In my previous post, I mentioned that it wasÂ *not* so obvious (both from a *physicalÂ *as well as from aÂ *mathematicalÂ *point of view) to write the wavefunction for electron orbitals – which we denoted as Ď(* x*,

*t*), i.e. a function of

*two*variables (or four: one time coordinate and three space coordinates) –Â as the product of two

*other*functions in

*one*variable only.

[…] OK. The above sentence is difficult to read. Let me write in math. đ It isÂ *notÂ *so obvious to write Ď(* x*,

*t*) as:

Ď(* x*,

*t*) =

*e*

^{âiÂˇ(E/Ä§)Âˇt}ÂˇĎ(

*)*

**x**As I mentioned before, the physicists’ use of the same symbol (Ď, *psi*) for both the Ď(* x*,

*t*) and Ď(

*) function is quite confusing – because the two functions areÂ*

**x***veryÂ*different:

- Ď(
,**x***t*) is a complex-valued function of*two*Â (real)*Â*variables:and*x**t*. OrÂ four, I should say, because= (**x**Â*x*,*y*,*z*) – but it’s probably easier to think ofas oneÂ*x**vectorÂ*variable – aÂ*vector-valued argument*, so to speak. And then*t*is, of course, just aÂ*scalarÂ*variable. So… Well… A function of*twoÂ*variables: the position in space (), and time (*x**t*). - In contrast, Ď(
) is a**x***real-valuedÂ*function ofÂ*oneÂ*(vector) variable only:, so that’s the position in space only.*x*

Now you should cry foul, of course: Ď(* x*) is

*notÂ*necessarilyÂ real-valued. It

*mayÂ*be complex-valued. You’re right.Â You know the formula:Note the derivation of this formula involved a switch from Cartesian to polar coordinates here, so from

**= (**

*xÂ**x*,

*y*,

*z*) to

*= (*

**r**Â*r*, Î¸, Ď), and that the function is also a function of the twoÂ quantum numbers

*Â l*and

*m*now, i.e. the orbital angular momentum (

*l*) and its z-component (

*m*) respectively. In my previous post(s), I gave you the formulas for Y

*(Î¸, Ď) and F*

_{l,m}*(*

_{l,m}*r*) respectively. F

*(*

_{l,m}*r*) was a real-valued function alright, but the Y

*(Î¸, Ď) had that*

_{l,m}*e*

^{iÂˇmÂˇĎ}Â factor in it. So… Yes. You’re right: the Y

*(Î¸, Ď) function is real-valued if – and*

_{l,m}*onlyÂ*if –

*m*= 0, in which case

*e*

^{iÂˇmÂˇĎ}Â = 1.Â Let me copy the table from Feynman’s treatment of the topic once again:The P

_{l}*(cosÎ¸) functions are the so-called (associated) Legendre polynomials, and the formula for these functions is rather horrible:Don’t worry about it too much: just note the P*

^{m}

_{l}*(*

^{m}*cos*Î¸)Â is aÂ

*real-valuedÂ*function. The point is the following:theÂ Ď(

*,*

**x***t*) is a

*complex-valuedÂ*function because – andÂ

*onlyÂ*because – we multiply a

*real-valued*envelope function – which depends on

*positionÂ*only – with

*e*

^{âiÂˇ(E/Ä§)Âˇt}Âˇ

*e*

^{iÂˇmÂˇĎ}Â =

*e*

^{âiÂˇ[(E/Ä§)ÂˇtÂ âÂ }

^{mÂˇĎ]}.

[…]

Please read the above once again and – more importantly – * think about it for a while*. đ You’ll have to agree with the following:

- As mentioned in my previous post,Â the
*e*^{iÂˇmÂˇĎ}Â factor just gives us phase shift: just aÂ re-set of our zero point for measuring time, so to speak, and the whole*e*^{âiÂˇ[(E/Ä§)ÂˇtÂ âÂ }^{mÂˇĎ]}Â factor just disappears when weâre calculating probabilities. - The envelope function gives us the basic amplitude – in theÂ
*classicalÂ*sense of the word:Â the*maximum*displacement fromÂ theÂ zeroÂ value. And so it’s that*e*^{âiÂˇ[(E/Ä§)ÂˇtÂ âÂ }^{mÂˇĎ]}Â that ensures the whole expression somehow captures the*energy*Â of the oscillation.

Let’s first look at the envelope function again. Let me copy the illustration forÂ *n* = 5 and *lÂ *= 2 from aÂ *Wikimedia Commons*Â article.Â Note the symmetry planes:

- Any plane containing theÂ
*z-*axis is a symmetry plane – like a mirror in which we can reflect one half of theÂ*shape*to get the other half. [Note that I am talking theÂ*shapeÂ*only here. Forget about the colors for a while – as these reflect the*complex*phase of the wavefunction.] - Likewise, the plane containingÂ
*bothÂ*the*x*– and the*y*-axis is a symmetry plane as well.

The first symmetry plane – or symmetryÂ *line*, really (i.e. theÂ *z*-axis) – should not surprise us, because the azimuthal angle Ď is conspicuously absent in the formula for our envelope function if, as we are doing in this article here, we merge theÂ *e*^{iÂˇmÂˇĎ}Â factor with the *e*^{âiÂˇ(E/Ä§)Âˇt}, so it’s just part and parcel of what the author of the illustrations above refers to as the ‘complex phase’ of our wavefunction.Â OK. Clear enough – I hope. đ But why is theÂ the *xy*-plane a symmetry plane too? We need to look at that monstrous formula for the P_{l}* ^{m}*(

*cos*Î¸) function here: just note the

*cos*Î¸ argument in it is being

*squaredÂ*before it’s used in all of the other manipulation. Now, we know that

*cos*Î¸ =

*sin*(Ď/2Â âÂ Î¸). So we can define someÂ

*newÂ*angle – let’s just call it Îą – which is measured in the way we’re used to measuring angle, which is

*notÂ*from the

*z*-axis but from the

*xy*-plane. So we write:

*cos*Î¸ =

*sin*(Ď/2Â âÂ Î¸) =

*sin*Îą. The illustration below may or may not help you to see what we’re doing here.So… To make a long story short, we can substitute the

*cos*Î¸ argument in the P

_{l}*(*

^{m}*cos*Î¸) function for

*sin*Îą =

*sin*(Ď/2Â âÂ Î¸). Now, if the

*xy*-plane is a symmetry plane, then we must find the same value for P

_{l}*(*

^{m}*sin*Îą) and P

_{l}*[*

^{m}*sin*(âÎą)]. Now, that’s not obvious, because

*sin*(âÎą) = â

*sin*Îą â Â

*sin*Îą. However, because the argument in that P

_{l}*(*

^{m}*x*) function is being squared before any other operation (like subtracting 1 and exponentiating the result), it is OK: [â

*sin*Îą]

^{2}Â = [

*sin*Îą]

^{2Â }=Â

*sin*

^{2}Îą. […] OK, I am sure the geeks amongst my readers will be able to explain this more rigorously. In fact, I

*hope*they’ll have a look at it, because there’s also that

*d*

^{l+m}/

*dx*

^{l+m}Â operator, and so you should check what happens with the minus sign there. đ

[…] Well… By now, you’re probably totally lost, but the fact of the matter is that we’ve got a beautiful result here. Let me highlight the most significant results:

- AÂ
*definiteÂ*energy state of a hydrogen atom (or of an electron orbiting around some nucleus, I should say) appears to us as some beautifully shaped orbital – an*envelopeÂ*function in three dimensions, really – whichÂ has the*z*-axis – i.e. the vertical axis – as a symmetry*line*and the xy-plane as a symmetry*plane*. - The
*e*^{âiÂˇ[(E/Ä§)ÂˇtÂ âÂ }^{mÂˇĎ]}Â factor gives us the oscillation*within*the envelope function. As such, it’s this factor that, somehow,Â captures the*energy*Â of the oscillation.

It’s worth thinking about this. Look at the geometry of the situation again – as depicted below. We’re looking at the situation along the *x*-axis, in the direction of the origin, which is the nucleus of our atom.

The *e*^{iÂˇmÂˇĎ}Â factor just gives us phase shift: just aÂ re-set of our zero point for measuring time, so to speak. Interesting, weird – but probably less relevant than the *e*^{âiÂˇ[(E/Ä§)Âˇt}Â factor, which gives us the two-dimensional oscillation that captures the energy of the state.

Now, the obvious question is: the oscillation of *what*, exactly? I am not quite sure but – as I explained in my *Deep BlueÂ *page – the real and imaginary part of our wavefunction are really like the electric and magnetic field vector of an oscillating electromagnetic field (think of electromagnetic *radiation* – if that makes it easier). Hence, just like the electric and magnetic field vector represent some rapidly changing *forceÂ *on a unit charge, the real and imaginary part of our wavefunction must also represent some rapidly changingÂ *forceÂ *on… Well… I am not quite sure on what though. The unit charge is usually defined as the charge of a *proton *– rather than an electron – but then forces act on some mass, right? And the *massÂ *of a proton is hugely different from the mass of an electron. The same electric (or magnetic) force will, therefore, give a hugely different acceleration to both.

So… Well… My guts instinct tells me the real and imaginary part of our wavefunction just represent, somehow, a rapidly changing force on some *unit *ofÂ mass, but then I am not sure how to define that unit right now (it’s probably *notÂ *the kilogram!).

Now, there is another thing we should note here: we’re actually sort of de-constructing a *rotationÂ *(look at the illustration above once again) in two linearly oscillating vectors – one along the *z*-axis and the other along the *y*-axis.Â Hence, in essence, we’re actually talking about something that’s *spinning.Â *In other words, we’re actually talking someÂ *torqueÂ *around the *x*-axis. In what direction? I think that shouldn’t matter – that we can write E or âE, in other words, but… Well… I need to explore this further – as should you! đ

Let me just add one more note on the *e*^{iÂˇmÂˇĎ}Â factor. It sort of defines the *geometryÂ *of the complex phase itself. Look at the illustration below. Click on it to enlarge it if necessary – or, better still, visit the magnificent Wikimedia Commons article from which I get these illustrations. These are the orbitals *nÂ *= 4 and *lÂ *= 3. Look at the red hues in particular – or the blue – whatever: focus on one color only, and see how how – for *m*Â *= *Âą1, we’ve got one appearance of that color only. For *m*Â *= *Âą1, the same color appears at two ends of the ‘tubes’ – or *toriÂ *(plural of *torus*), I should say – just to sound more professional. đ For *m*Â *= *Âą2, the torus consists of *three* parts – or, in mathematical terms, we’d say the order of its *rotational symmetry*Â is equal to 3.Â Check that Wikimedia Commons article for higher values ofÂ *nÂ *andÂ *l*: the shapes become very convoluted, but the observation holds. đ

Have fun thinking all of this through for yourself – and please do look at those symmetries in particular. đ

**Post scriptum**: You should do some thinking on whether or not theseÂ *mÂ *=Â Âą1, Âą2,…, Âą*lÂ *orbitals are really different. As I mentioned above, a phase difference is just what it is: a re-set of the *t* = 0 point. Nothing more, nothing less. So… Well… As far as I am concerned, that’s notÂ aÂ *realÂ *difference, is it? đ As with other stuff, I’ll let you think about this for yourself.

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