What, really, is the Variational technique?

[ognik]

I have come across mention of using 'the variational technique' for solving the 1-D Schrödinger (and I assume many more eqtns). I really don't understand what the variational technique is, not getting anywhere fast by googling, can someone walk me through the basics please? I gather it is used for solving ODEs for example ...

I think it has to with the variational principle which is (Wiki) 'general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions'. An example (also from Wiki) goes: "What is the shape of a chain suspended at both ends?" - we can use the variational principle that the shape must minimize the gravitational potential energy.

In other words it will hang down? Clearly it must be more interesting than this example suggests to me, would appreciate some insights & intuitions. Maybe even an example of a function that can be found to minimize values w.r.t. some well known function? Thanks

 

[ZapperZ]

Back up a bit.

You went straight into variational technique for Schrödinger equation. Have you not done such a thing in classical mechanics first before jumping into quantum mechanics? Have you not done Lagrangian/Hamiltonian mechanics?

Mary Boas's text "Mathematical Methods in the Physical Sciences" has a whole chapter on Calculus of Variation and the Least Action principle. That chapter alone is worth getting the book.

Zz.

 

[ognik]

You went straight into variational technique for Schrödinger equation. Have you not done such a thing in classical mechanics first before jumping into quantum mechanics? Have you not done Lagrangian/Hamiltonian mechanics?

I did my physics over 20 years ago, we didn't cover Lagrangian/Hamiltonian mechanics, it is on my to do list, but it's quite a hefty topic and I have course deadlines ...

Mary Boas's text "Mathematical Methods in the Physical Sciences" has a whole chapter on Calculus of Variation and the Least Action principle. That chapter alone is worth getting the book.

I've kind of given up work to study, so funds ... could someone point me at an online resource - a nice 'for dummies' type starter?