- #1
SemM
Gold Member
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Hi, I have been looking in various text about how to find an admissible solution to the Schrödinger eqn in one dim. in the harmonic oscillator model. As in MQM, the solutions to this are said to be ##Ae^{ikx}+Be^{-ikx}##, which are then said to be not admissible. The book then goes straigtht to the Hermite polynomials as solutions.
In Bohms Quantum Theory, he splits up the Schrödinger eqn into its factorial form, and uses the two factorial forms to show how the raising and lowering operators are used to develop functions of higher order (and lower order).. Here he goes straight to show an example, ##\psi=e^{-x^2}##. He elaborates further on the operators, the generating function and so forth.
In Pauling and Wilsons Quatum Mechanics, on p 69, they write, similar to MQM, the Schrödinger eqn for the harmonic oscillator, and then write:
"This equation is satisfied asymptoticallly by the exponential functions:
\begin{equation}
\psi = e^{\pm \alpha/2 x^2}
\end{equation}
and no way is given on how this is derived, which is surprising, because in all Math books of graduate level, some method for showing how to derive the solution from the original is always given, i.e such as the method of ##Ae^{\lambda_1x}+Be^{\lambda_2x}## for the ODE form ##ay''+by'+cy=0##, where ##\lambda = [-b\pm\sqrt{b^2-4ac}]/2##.
As one who has recently studied ODEs in both Greenbergs Math, and Kreyszig Adv Eng Math, I was expecting that physicians in all these books, would list up a method for deriving a solution, and I am surprised that always the same example recurs (the form given above and in Bohm and Pauling), and not other versions of it or of similar admissible functions. Especially, when it is known that ODEs have many solutions, and in quantum chemistry knowing several solutions is a great start for DFT methods to calculate the properties of electrons (through the cycling method).
1. Are there really so few admissible examples in literature, and surprisingly, no mathematical method to derive these solns. (i.e. the Bernoulli eqn. method for solving ODEs, or the method for solving the Riccati eqn. etc. etc.)?
2. Can someone provide a method for solving the S.eq. and get the soln. given above, (Pauling and Wilson , and Bohm)?Note that another similar post has been posted previously, but this post has more references, the same example, and even more important, an uttering on whether someone can provide a mathematic method to solve S.Eq. and get the admissible form given above?
This is a question about method, not about why the given soln. results.
Thanks
In Bohms Quantum Theory, he splits up the Schrödinger eqn into its factorial form, and uses the two factorial forms to show how the raising and lowering operators are used to develop functions of higher order (and lower order).. Here he goes straight to show an example, ##\psi=e^{-x^2}##. He elaborates further on the operators, the generating function and so forth.
In Pauling and Wilsons Quatum Mechanics, on p 69, they write, similar to MQM, the Schrödinger eqn for the harmonic oscillator, and then write:
"This equation is satisfied asymptoticallly by the exponential functions:
\begin{equation}
\psi = e^{\pm \alpha/2 x^2}
\end{equation}
and no way is given on how this is derived, which is surprising, because in all Math books of graduate level, some method for showing how to derive the solution from the original is always given, i.e such as the method of ##Ae^{\lambda_1x}+Be^{\lambda_2x}## for the ODE form ##ay''+by'+cy=0##, where ##\lambda = [-b\pm\sqrt{b^2-4ac}]/2##.
As one who has recently studied ODEs in both Greenbergs Math, and Kreyszig Adv Eng Math, I was expecting that physicians in all these books, would list up a method for deriving a solution, and I am surprised that always the same example recurs (the form given above and in Bohm and Pauling), and not other versions of it or of similar admissible functions. Especially, when it is known that ODEs have many solutions, and in quantum chemistry knowing several solutions is a great start for DFT methods to calculate the properties of electrons (through the cycling method).
1. Are there really so few admissible examples in literature, and surprisingly, no mathematical method to derive these solns. (i.e. the Bernoulli eqn. method for solving ODEs, or the method for solving the Riccati eqn. etc. etc.)?
2. Can someone provide a method for solving the S.eq. and get the soln. given above, (Pauling and Wilson , and Bohm)?Note that another similar post has been posted previously, but this post has more references, the same example, and even more important, an uttering on whether someone can provide a mathematic method to solve S.Eq. and get the admissible form given above?
This is a question about method, not about why the given soln. results.
Thanks
Last edited: