The Hamiltonian

$H_{\psi }=E_{\psi }$

1-D: $E_{TOTAL}\psi (x)=E_{k}\psi (x)+V(x)\psi(x)=-\frac{h^{2}}{8\pi ^{2}m}\frac{d^{2}\psi(x)}{dx^{2}}+V(x)\psi(x)$

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Jonathan Batac - 2D
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Joined: Wed Sep 30, 2020 10:03 pm

The Hamiltonian

What is the significance of the Hamiltonian in Schrodinger's equation? I understand that E is the energy of the election and the psi symbol represents the wave function (or the orbital) of the electron, but how does the Hamiltonian relate to any one of these variables?

KatarinaReid_3H
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Re: The Hamiltonian

The hamiltonian means take the second derivative of the number given. In Dr. Lavelle's lecture he said that for example d/dx sin is cos, d/dx cos is sinx so taking the double derivative of the sine and cosine trig functions causes you to end with the same initial value. I repeats in a cycle.

Leo Naylor 2F
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Re: The Hamiltonian

Good explanation, Katarina. I would just like to clarify that the derivative of sin(x) is cos(x), but the derivative of cos(x) is -sin(x). The derivative of -sin(x) is -cos(x) and the derivative of -cos(x) is sin(x). It is a cycle, but it repeats every four derivatives instead of every two.

Kelly Yun 2I
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Joined: Wed Sep 30, 2020 10:03 pm

Re: The Hamiltonian

Thank you for your explanations--I also had a question about the hamiltonian. I understand that it is a double derivative but I searched it up and read that it was the total energy(?) which I don't understand. Is the hamiltonian just a mathematical symbol or is it representing the energy?

Leo Naylor 2F
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Re: The Hamiltonian

In the lecture about Schrodinger's wave function equation, Professor Lavelle mentioned that if one "operates a change on ψ(x,y,z), it would equal E*ψ(x,y,z)." The change operation was represented as H, meaning H*ψ = E*ψ. Therefore, H would be equal to the energy of the system. That being said, the math here involves a three dimensional sine wave, which is far beyond my understanding and the understanding required for this course.

Xavier Herrera 3H
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Re: The Hamiltonian

In the equation H*wave function=E*wave function, the wave function stays the same on both sides of the equal sign, so H and E must be equal to each other. E represents the energy of the electron in an atom, so if you take the double derivative of the wave function, you get the energy of the electron.

Mackenzie Stockton 2H
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Joined: Wed Sep 30, 2020 10:11 pm

Re: The Hamiltonian

The Hamiltonian of the wave function is equal to the energy of the wave function. Since the wave function is sin theta/cos theta, the Hamiltonian is taking the double derivative of the wave function (sin theta), which gives you back sin theta. because of this fact, you are given back the wave function on the right side of the schrodinger equation, as well as the energy of this same wave function. I think that for this class, the importance of the Hamiltonian is that is is the double derivative of the wave function. Also, it is important to note that the wave function in this math equation is what gives us the specific orbitals that are currently discussion in class.

Jerry_T
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Re: The Hamiltonian

The Hamiltonian is just a specific word for double derivative in this context.

Andre Fabian 1F
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Re: The Hamiltonian

In this context, a "Hamiltonian" represents a double derivative. Thus, the Schrodinger equation represents that the double derivative of the wave function (orbital) representing e- is equal to the Energy of the wave function (orbital) representing e-.

Hope this helps!
Andre

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