I already knew the physical significance, and for each of the steps, I could relate them to their physical significance, so I kinda understood it better. I dunno how would I get this mathematical stuff without some physical implications. Instead of making it all mathematical, I think it would be great if you would mention each element's or step's physical significance too. Otherwise, you are great and a life saver, man.
Yeah, it starts getting very tricky at this point in the series, because the approximation methods are very abstract to begin with. I think it makes a lot more sense when you examine concrete examples (like the next video, 8.8) where you see how the input parameters make their way into the output solution, and how turning those knobs affects the result.
One way would be to assume real functions and not deal with the headaches that come with generalizing to complex function values. Even if we don't, it still works out with a little more math. It just happens to be more complicated and nuanced than I'd like to cover in a can be expressed in the basis set of zeroth-order basis functions (|n0,1>, |n0,2>, |n0,3>, etc.). Since these are eigenfunctions of a Hermitian operator (the zeroth-order Hamiltonian), we know that they are orthonormal. Since the ket vector in this overlap integral is |n0>, all components of |n1> which belong to other zeroth-order eigenfunctions go to zero (satisfying the real number constraint). The only non-zero component left over is (c0^2), and is a real number. equals this integral plus a sum of other components which are all zero, thus is real, and thus so is , therefore = even in the most general complex case.
Hi Vanessa. Algebraically, this is an identity operation similar to multiplying both sides by the same number. We multiply all terms on both sides by psi_star and integrate, leaving the equality in tact. The reason to do this is the same reason we do so in all such algebraic transformations: to refactor the equation into a more convenient and solvable form. After some factoring over the next two lines, 3 out of 4 integrals either end up being 0 or 1, giving us our final desired result of the first-order energy in terms of an integral which we can solve for a given wavefunction and perturbation operator.
I luv u man.! You cleared my all thoughts in just one video.!! May you get more success in life...
How < n0 | n1> is rewritten as < n1 | n0> in the line 4 of the right paragraph?
Appreciate your work!
I already knew the physical significance, and for each of the steps, I could relate them to their physical significance, so I kinda understood it better. I dunno how would I get this mathematical stuff without some physical implications. Instead of making it all mathematical, I think it would be great if you would mention each element's or step's physical significance too. Otherwise, you are great and a life saver, man.
Yeah, it starts getting very tricky at this point in the series, because the approximation methods are very abstract to begin with. I think it makes a lot more sense when you examine concrete examples (like the next video, 8.8) where you see how the input parameters make their way into the output solution, and how turning those knobs affects the result.
On the 2nd half of the page, from the magenta to the yellow line, how did you change to ? Do we assume they are real functions? If so, why?
One way would be to assume real functions and not deal with the headaches that come with generalizing to complex function values. Even if we don't, it still works out with a little more math. It just happens to be more complicated and nuanced than I'd like to cover in a can be expressed in the basis set of zeroth-order basis functions (|n0,1>, |n0,2>, |n0,3>, etc.). Since these are eigenfunctions of a Hermitian operator (the zeroth-order Hamiltonian), we know that they are orthonormal. Since the ket vector in this overlap integral is |n0>, all components of |n1> which belong to other zeroth-order eigenfunctions go to zero (satisfying the real number constraint). The only non-zero component left over is (c0^2), and is a real number. equals this integral plus a sum of other components which are all zero, thus is real, and thus so is , therefore = even in the most general complex case.
TMP Chem Thank you for answering!
Why did you multiply by Psi(star) after you changed the equation into Dirac?
Hi Vanessa. Algebraically, this is an identity operation similar to multiplying both sides by the same number. We multiply all terms on both sides by psi_star and integrate, leaving the equality in tact. The reason to do this is the same reason we do so in all such algebraic transformations: to refactor the equation into a more convenient and solvable form. After some factoring over the next two lines, 3 out of 4 integrals either end up being 0 or 1, giving us our final desired result of the first-order energy in terms of an integral which we can solve for a given wavefunction and perturbation operator.
Great! Thank you!
THANK YOU! 💕💕