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Model Spectroscopy and Eigenvectors/モデル分光法と固有ベクトル

Model Spectroscopy and Eigenvectors/モデル分光法と固有ベクトル
We are here again with Assistant Professor James Whitfield from Dartmouth College to talk about quantum chemistry. James, thanks for being back with us. I’m glad to be back. Let’s talk about quantum chemistry algorithms. Yes, so quantum chemistry algorithms are really about… I like to think of it as model spectroscopy. Spectroscopy is perhaps a fancy word, but really in the course you’ve already covered Fourier transforms. When I think of Fourier transform, as you hit a tuning fork, you Fourier transform to find out what that frequency is. The same thing happens to molecules.
You take a molecule, you hit it with light or even thermal energy and then it vibrates or responds in the case of electronic states not vibrations, but the response of light and then you can read out whether the light has been observed or the light as been transmitted. Whether it goes through, it is just like tuning fork analogy. You want to know what frequencies this thing responds to. And with quantum computation, the idea is to actually make model quantum systems, right. So we quantum system we can control on a quantum computer and with that quantum computer we can then take the quantum computer and map the system we’re interested in.
Water, protein or a metal and map to this quantum computational device, use that computational device to then simulate the system we’re interested in and then we can hit the system perform the Fournier transform and then understand the frequencies, the energies that are associated with that system. So tell me what an eigen vector and an eigen value is – and in this context in particular. This is a great place to kind of refer to some of the earlier lessons that you already covered so with Fourier transforms and with linear algebra systems.
If we think about a stationary wave, a wave that is just going up and down (but not necessarily two of them just one), and you take the amplitude staying constant in time and the frequency staying constant in time, well then that wave is effectively stationary versus shaking the rope and watching that wave propagate. You can see something a disturbance moving through the system. In this case disturbance everywhere and is stationary it is not moving anymore. This is called the stationary state. The Hamiltonian your energy operator, it is going to tell you about the kinetic and the potential energy and how these two things combined has stationary states and these are the eigenvectors and eigenvalues.
We just think of, when you think about eigenvectors and eigenvalues is that “eigen” is a word that means ‘own’ in German. I didn’t know that. Mean as in ‘belongs to you.’ Yeah. “Eigenworth” would be like your self-worth for instance. So it is really something belonging to you. And so when you think of these of operators they we’re going to have things that really, really dictate a large part of their description. These would be vectors and values corresponding those vectors and I think a nice analogy for an eigenvector is to think of the Earth rotating. I you have the Earth rotating and you stand at some point that is not on the axis of rotation, well, you are going to move.
When you stand on the axis of rotation you think you are not going to move anymore. This is the axis of rotation straight up and you can imagine rotating around, so if I take some vector here it just going to rotate and precess around. When I take a vector here, as I rotate this vector nothing is going to happen because rotating it, it is still the same vector. So this point then is the North Pole? Right! From the center to the North Pole, so it shouldn’t be a point it should be a vector. Hamiltonian is a linear operator meaning whatever you said earlier about the linear operators, but essentially it is going to take vectors and transform into new vectors.
If it transfers one vector to the same vector then that is called an eigenvector. Now it may change the length of that vector and the change in length is called the eigenvalue. So it takes a vector and just distorts it a little bit, but does not change its direction, just the length and that change in length is eigenvalue and the vector that doesn’t change when you apply as operator is the eigenvector. Great. So how do eigenvector and eigenvalue figure into quantum chemistry? What do they tell us about the system? For Hamiltonians, the eigenvalues in the Hamiltonian are the sum of the kinetic plus the potential energies. This tells you about…
You can think that if we had an apple on the table, we knock it off the table and it falls down. When it hits the bottom it cannot go any further because now it is on the ground. This would be the ground state. If you have to excited states, which have more energy and they can lose that energy (produce a photon for instance, and you would see it), or absorb a photon and go to the higher energy state, but only at quantized values, which is really where quantum mechanics gets its name. Right,
Darmouth大学のJames Whitfield教授は、量子化学アルゴリズムの第一人者です。 彼はこれらのアルゴリズムを”モデル分光法”として考え、重要な周波数とエネルギーを発見しました。

固有ベクトル固有値は、それらの周波数とエネルギーを定義するのに役立ちます。 オペレータの固有状態は、システムのオペレータの特徴を決定づけます。

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