Page 1 ( Victor Galitski )

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So, here is the Schrodinger equation, once again we're going to see it very often in this course. And I'm also showing here , a part of the first page of the original paper by Schrodinger, published in December of 1926.
And it's actually very interesting how Schrodinger came up with this work and this equation.

Apparently the story started back in 1925 or so, when he was working under Debye, and Debye had just read a paper about de Broglie, where de Broglie was introducing his wave particle duality ideas.
So, he got interested in these ideas and suggested Schrodinger to give a seminar on de Broglie's work. So apparently, Schrodinger actually dismissed this at first, saying that he didn't even want to think about such a silly theory.
But he had to give in because well, Debye was effectively his supervisor. And so, he was looking into de Broglie's work, trying to present it in a more mathematically sophisticated form.
And in doing so, he came up with the Schrodinger equation essentially, as we now know it. And which brought him the worldwide recognition and a Nobel Prize in Physics in 1933.

Another important thing that Schrodinger did was that he used his equation to solve a very important problem of the charged particle and the cool-down potential, which essentially describes a quantum hydrogen atom
and he found the energy level structure which is which was consistent with de Broglie's atom.

In quantum mechanics the momentum is an operator so velocity which is momentum divided by m is also an operator.
The final result which connects the change in the probability of finding a particle in the volume v with the flux of a certain probability current flowing through the surface.

 

In the shallow quantum well, in one dimension, or in the corresponding delta potential well, there exists one, and only one, bound state with this energy.