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1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacket.

KarlBogha
Amethyst

1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacket.

QM Demystified A Self Teaching Guide by David McMahon. I purchased this
book in 2007. Read chapter 1 and 2 in that year. Did not get past chapter 2, but did complete chapter 2. May had gotten busy with other things, most probably found the subject matter incomprehensible, difficult, tough, .... This book made it possible for me to understand the basic mathematics required for the topic. I revisited it again in 2019 and 2020, as you can see.

Had it not been for David McMahon's book, Schrodinger Equation may not been a subject matter I could understand. I admit I may still be incomplete on its understanding. So there maybe mistakes here that you can easily spot. Apologies in advance.

Example were needed to supplement the theory, this is why this book was suitable for me. There are a few examples from other textbook(s) in the study notes.

Its mainly the mathematics technique required and applied to understand some of the undergraduate Schrodinger Equation subject matter.

Approximately 85 pages. Tried to make it as step by step as possible.

21 examples in all.

You may need to zoom in at a higher magnification for the subscripts.

Any errors and omissions apologies in advance.

----------------------------------------------------------------

From Wavefunctions and Wavepacket To Schrodinger Equation.

Intermediate Level.

I found this topic loosely attempted, maybe a lack of full appreciation where I was not confident.

I had the steps thru once. The assumptions their impact, and the full picture on the wavepacket were not firm. Not that they are now you can or may proof that.

If you had a course in communication theory in electrical engineering or similar subject, maybe you sensed the same hurdle I did in that subject, similar here.

Its here where the physicists Mahesh Jain, who wrote QM For UGs, makes his case to end one kind of material and leap into another. The another being the Schrodinger Equation.

His approach on how to show the wavefunctions superpositioned to form the wavepacket. The phase velocity to the group velocity. Putting it another way, if correct, to show how the phase velocity is 'embedded' in the group velocity.  Embedded sounds more engineering like but it maybe the wrong expression to use here. This is the subject matter here. It comes just before Schrodinger Equation. 

No solved examples.

Just to show how to get from the 'wavefunctions and phase velocity' to the 'wavepacket and group velocity'.

I call it a ROUGH DRAFT because I am not prepared to take on that role as a subject matter expert on this topic. You may try and are welcome.

Content based on Mahesh Jain's textbook titled Quantum Mechanics For Undergraduates. Publisher PHI.

Any errors and omissions apologies in advance.

N/A.
22 REPLIES 22

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Hi,

A few years ago, I published QM (https://community.ptc.com/t5/PTC-Mathcad/QM/mp/589937/thread-id/184311?attachment-id=72746) and QE (https: // sejnu66972 .i.lithium.com / t5 / image / serverpage / image-id / 22166iC18B16F3DD134D74 / image-size / large? v = 1.0 & px = 999) as a feasibility study of the general formalism with the use of MATHCAD 15. But it is impossible given the difficulties in using ket and bra.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

File not available to the link provided.

Schrodinger I need to be on it regular to keep up with it.

In my case I have to resort to the textbook and notes to get back to what I know.

Quantum Physics is a good learning subject, and solar PV make a good case for the engineering side of things.

Its in electronics yes but the PV is more closer a physical object to the micro-electronics under the micro-scope.

 

Regards,

Karl.

N/A.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Saphire II,

 

You work is very strong from the university environment. The early topics you have in the pdf file assume you are familiar with the UG content of introduction to modern physics. Its very hieroglyphic heavy with symbols from Mathematics language. It does seem intimidating it is intimidating, maybe this you should have broke down with examples for an introductory presentation, otherwise for those who know the prerequisite of the subject matter then that's fine. Personally it turns me off. Thanks.

N/A.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Hi,

Thanks to Vladimir for posting the correct address. But I want to clarify that I am the author of the two files (I wrote them myself). Rereading them, after a long time, I must say that there is something questionable.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

You are saying there is something questionable in the 2 files Saphire II posted, that's fine. You read them and found something questionable. Good. You're learning. 

N/A.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Hi,

They are drafts. They serve to show that with Mathcad you can go beyond what it offers, albeit with limitations, and not to make theory. Although I have passed seventy years by a few years, it is known that there is always something to learn.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Maybe I got you now MFra!

Maybe, so Saphire II is a designation.

I just made member.

I was here in 2016 been busy later, did not make it here, had a few files I sensed may help UG students.

 

 

 

Regards,

Karl.

N/A.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Karl,

I think you may have become confused over who wrote the worksheets under discussion.

VladimirN posted the links to the files. He is not the worksheets' author. (BTW, "Sapphire" is Vladimir's "Community Rank", not his user name)

-MFra- (rank "Topaz") is the author of the worksheets.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Makes a big difference. Thanks.

N/A.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

No worries, Karl.

 

Stuart

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Stuart,

Why did I get several responses from 'generalist' rather than physics majors or EE's?

 

I got a thread going on my Schrodinger Equation.....but none of the respondents are related to this interest.

 

Next time I may have to filter my response.

 

Too many Chiefs Not Enough Chinese.

Thanks. 

N/A.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke


@KarlBogha wrote:

Stuart,

Why did I get several responses from 'generalist' rather than physics majors or EE's?

 

I got a thread going on my Schrodinger Equation.....but none of the respondents are related to this interest.

 

Next time I may have to filter my response.

 

Too many Chiefs Not Enough Chinese.

 


It won't do you any  good, Karl.  We'll track you down no matter where you hide in the Community, and we hunt in packs.  🐺

 

It's just one of life's cruel mysteries that it seems to be the generalists who tend to answer questions.  The slightest sniff of an interesting problem, or opportunity to educate, and there they are.  

 

(OTOH, you may find that several of the generalists do have backgrounds in physics or EE - my degree's in astrophysics and I've worked on radar and radio, but I'm a systems person by temperament and tend to poke my nose into anything)

 

Cheers,

 

Stuart

 


 

If you wait long enough, the generalists will all drift away to more interesting climes.  One of the sad things about Mathcad and this Community is that many of main respondents are long-time Mathcad users, most of whom cut their teeth on Mathcad way back in the days when Mathsoft owned the product.   However, Prime is somewhat of a disappointment to many of them and several of the top contributors have left or are rarely seen walking the hallowed halls of the Community.   Worse, there aren't many newcomers filling out the gaps in the pack, and the Community seems to be a lot quieter than the old Mathsoft Collaboratory.  IMHO, PTC Marketing ought to be out there beating the Mathcad drum a little more loudly than it does, otherwise fresh meat for the pack is going to be in awfully short supply. 

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

" IMHO, PTC Marketing ought to be out there beating the Mathcad drum a little more loudly than it does, otherwise fresh meat for the pack is going to be in awfully short supply."

 

Well spoken!  (Written?)  However, my observation for some time now is that PTC's intent for Mathcad is to be an embedded calculator for CREO.  Those of us who used Mathcad for what it does (a more easily read, less program language required, competitor for MATLAB?) are an embarrassment, since we keep pointing out that the original version (how old now?) can still do things that Prime cannot; and they are waiting patiently for us to go away.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

You are right about beating the drum and selling more Mathcad. For that matter Prime too, if the older version is not supported. I heard about this when I first got on Prime, then Mathcad 15 was not available and may been the last version of significance, to most of your group, before Prime. 

 

Thanks.

 

 

N/A.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

"I heard about this when I first got on Prime, then Mathcad 15 was not available and may been the last version of significance, to most of your group, before Prime."

 

Sorry!  Mathcad 15 (which you must have installed so Prime can read and convert earlier versions of Mathcad) is still available and runs on the same license as Prime.  And it can still do things that Prime (at version 6.0) won't do.

 

As I've said before:  "Pay for the tail, get the dog for free!"

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke


@KarlBogha wrote:

You are right about beating the drum and selling more Mathcad. For that matter Prime too, if the older version is not supported. I heard about this when I first got on Prime, then Mathcad 15 was not available and may been the last version of significance, to most of your group, before Prime. 


Believe it or not, quite a few of us old-timers had high hopes when PTC took over Mathcad.    Several of us wanted far more capability in Mathcad to make it into a truly smart mathematical whiteboard / document production tool.   Unfortunately, that wasn't what we got - we got Prime.    I don't want to get into Prime-bashing, but suffice it to say that the reason so many old Mathcadders don't particularly Prime has got nothing to do with it being 'different' or 'new', but does have a lot to do with what Prime doesn't do.   Don't get me wrong, there have been some positives (particularly in array input and reversion to dynamic 'typing'(*)), but there is still so much missing.

 

Stuart

 

(*) I like functional programming and can work with a properly designed static type system, but the Mathcad 12..15 versions of static typing didn't fit well with Mathcad's way of working.   It might not have been so bad if Mathcad had fully supported functional programming (eg, proper recursive support and laziness), but it didn't.  Nor was there a proper "educational" program to help users understand what it was all about - if there was, we might not have lost partial application and I'm *really* missing that feature in Express.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke


@Fred_Kohlhepp wrote:

" IMHO, PTC Marketing ought to be out there beating the Mathcad drum a little more loudly than it does, otherwise fresh meat for the pack is going to be in awfully short supply."

 

Well spoken!  (Written?)  However, my observation for some time now is that PTC's intent for Mathcad is to be an embedded calculator for CREO.  Those of us who used Mathcad for what it does (a more easily read, less program language required, competitor for MATLAB?) are an embarrassment, since we keep pointing out that the original version (how old now?) can still do things that Prime cannot; and they are waiting patiently for us to go away.


Thanks, Fred.

 

Not sure if it counts as serendipity, but I was talking about this very thing to a few people on the PTC/User committee (only because I'd asked why no Mathcad presence at LiveWorx - Heaven forfend that reality would put me on a committee!). 

 

What people want from Mathcad is an interesting topic.   I'd long had a vision of Mathcad being a smart whiteboard that allowed people to develop an idea from the virtual back-of-a-beermat stage to the TeX/ LaTeX final report.  I didn't see Mathcad so much as a competitor to Matlab but rather as an adjunct to Matlab and other programming systems that covers the initial design and development stages - sort of a Jupyter on steroids.  For example, I've done the theoretical development for a simulation in Mathcad, implemented it in Simulink/Matlab, and verified the simulation using Mathcad-generated test data (as it was in Mathsoft Mathcad, I could drive the simulation directly via a component - "Look, Ma! No hands!").  Additionally, there are a large number of cases where Mathcad can/could do the whole lifecycle.

 

I'd like to see some engagement with PTC on this matter, particularly extending the debate to those who *don't* have a support contract  - because they're the ones *not* buying Mathcad Prime.    Part of the long-standing problem with upgrades was that even back in the Mathsoft days many users felt even M15 didn't offer them anything above the capabilities they enjoyed in M6..M11, and M12 went down like a lead balloon with many users.   As many have pointed out, Prime's offering less doesn't seem to be a valid strategy for increasing Mathcad sales.   And yet there is so much that could be done to improve Mathcad and bring it to the point where people actually want to get the newest version. 

 

 

Cheers,

 

Stuart

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Pyramid Thinking - This Time My Way!

 

I see you coming from an advanced user background. That is a plus for Prime/Mathcad.

Lets see if we, you too Fred, can work thru the management thinking on this. Lets try the Pyramid, you got the base of the Pyramid its wide its got a large cross section, here its the entry level or beginner, most number of users. Lets catch them here, lets not sell them too much, they may react negatively. As you move up the pyramid to about to the centroid or balanced mid-point position, lets say the company got the best/optimum number of customers and sale. From the base point to the centroid the company got the best dollar return, this includes cost of development, advertising, wages, and maintenance. Now do we go further and entertain a small elite group that reaches to the peak of the pyramid or slow it down past the centroid? And other factors to consider the business market landscape.........

 

This maybe one way of looking at it. 

 

If you write a book you are welcome to use that Pyramid correlation I presented here, original, give me some credit if you remember otherwise no worries.  

 

 

 

Regards,

Karl.

N/A.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Stuart,

Interesting thoughts!

 

" I didn't see Mathcad so much as a competitor to Matlab but rather as an adjunct to Matlab and other programming systems"

I use(d) Mathcad as a stand-alone engineering tool to deal with whatever problem the boss dropped on my desk.  I was taught FORTRAN in college and transitioned to BASIC when industry began to use desktop computers as data acquisition systems.  I've been "exposed" to MATLAB and "C" programming (you can't work in industry today without some supervisor directing you to run somebody else's MATLAB routine); but  they never "took."  I liked that Mathcad looked like what I used to create with paper and pencil!  I liked that it kept track of units and let me know when the units didn't balance.  (Why are there extra kilograms when I wanted the answer in feet per second?)  And I discovered that I could do almost anything my younger MATLAB savvy coworkers could do, with units!

"I'd like to see some engagement with PTC on this matter, particularly extending the debate to those who *don't* have a support contract  - because they're the ones *not* buying Mathcad Prime."

The "one good thing," IMHO, that PTC has done for Mathcad is Prime Express.  (Shhh, don't say it too loud!):

  When you download your evaluation copy of Prime, and the 30 day trial license expires, Prime becomes "Express," and a lot of the high-powered stuff stops working--solve blocks and symbolics for two of the big ones.  There's also a few annoying minor things (the "mean" function doesn't work for example) like the add-in plotting and (I believe) the embedded EXCEL sheet.  I think PTC intended this to be a "reader," it will open without calculating and Prime sheet; this allows you to see the results of the full sheet as it was last calculated before it was saved--the plots, solve blocks and symbolics all can be seen.  If you recalculate it a lot of stuff suddenly turns red.

What is left is something fairly close to what Mathcad looked like when I first met it on a Macintosh computer in the mid 1980's.  It's a fairly useful tool.  It requires some ingenuity to get around the things that have been disabled, but there's significant capability there, without programming!

 

" As many have pointed out, Prime's offering less doesn't seem to be a valid strategy for increasing Mathcad sales."

I don't believe PTC (management) ever planned to continue Mathcad as a stand-alone.  The speed of development, and what's been developed and what's been left undeveloped point to the intent to drop Mathcad into the Creo suite as a calculation/analysis addition to Creo.

 

But they have given us a free fairly useful stand-alone program.

 

 

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

You are welcome to comment on what I got.

Will you look into my social security payments I need them in a few years time.

 

 

 

Regards,

Karl.

 

N/A.

Re: 1) Quantum Mechanics: Schrodinger Equation Some Worked Examples. 2) Wavefunction & Wavepacke

Good FYI. I came across one feature which I could not do in Prime, it may been the convolution feature, where there was a short time period setting where you see the curves overlap in motion, something like that. That feature is not in the student version or tail version.

 

Pay for the tail and get the dog for free. This is a new one its funny. Cat's don't get near the attention dogs get, they missed out on the tail too. 

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