## Hamiltonian

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

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

Rachel Kho Disc 2G
Posts: 55
Joined: Wed Sep 30, 2020 9:46 pm

### Hamiltonian

Can anyone explain why the Hamiltonian is a double derivative? I remember learning and understanding it during lecture but as I'm looking back over my lecture notes I'm forgetting what was meant by that.

Ayesha Aslam-Mir 3C
Posts: 77
Joined: Wed Sep 30, 2020 9:43 pm

### Re: Hamiltonian

The Hamiltonian refers to taking the double-derivative of the wave function; think of how the wave function is either a sine or cosine function. The double derivative of sine or cosine would just be sine or cosine but with a different sign. Schrodinger's equation is actually a differential equation that just helps us calculate the wave function and relation to energy, which I think is the most important thing to know about this equation.

Posts: 58
Joined: Wed Sep 30, 2020 10:01 pm

### Re: Hamiltonian

The schrodinger equation says that H*wave function = E*wave function. So the equation returns the same wave function put in. In this equation, wave equation equals sin(theta) plus some other complexities that we aren’t learning. So when we take the double derivate of sin(theta) we get sin(theta) again, which is just calculus. So this is what the equation does, so in this case the Hamiltonian is a double derivative. Hope this makes sense.

Danielle Goldwirth 3F
Posts: 62
Joined: Wed Sep 30, 2020 10:07 pm

### Re: Hamiltonian

Furthermore, psi, which we defined to be the sine function, has positive and negative phases a long with the nodes.
When this function is squared, however, it transforms into the cosine function which becomes positive everywhere and also has nodes (areas of zero probability).
Because psi^2 represents the probability of finding an electron, it is important in the equation along with the psi, which is actually in the formula. So it is important to have the Hamiltonian, which can act somewhat recursively, to bring back the original math/wave function.