When the last video we showed why the variation on principal is true, namely why the expectation value of a Hamiltonian for some trial wave function that is not the true wave function of the system gives an upper bound for the ground state energy of that system. In this video we'll go through some tips and tricks for choosing a good trial wave function. And what I mean by that is one for which this ratio gets you as close as possible to the true ground state energy of the system. One of the tricks that is usually employed is the trial wave function is typically written in terms of one or more parameters that we'll call alpha 1, alpha 2, etc. which can be varied to minimize this ratio. So for some trial wave function that depends on some parameters alpha 1 and alpha 2, we calculate this ratio. This gives us an estimate for the energy in terms of these parameters. And then once we have that we want to minimize this estimated energy for each one of these parameters. And what that entails is for each one of those parameters the partial derivative with respect to alpha 1, alpha 2, etc. has to be equal to zero. Given some solutions to this minimization, so we'll call that alpha 1 bar, alpha 2 bar, etc. We plug that back into our estimated energy over here. So this is now alpha 1 bar, alpha 2 bar, etc. This will give the lowest possible energy for this trial wave function. So we're trying to optimize or in this case minimize the energy of a given trial wave function. And then the next video will will give an example of how this is done in practice. There are a number of other considerations that are usually taken in choosing trial wave function in practice. So the first one is we often have a sense of what the true wave function should look like at least qualitatively. This is typically based on symmetry properties or boundary conditions. The trial wave function is then chosen to have the same expected properties that we expect for the true wave function. And a simple example of that is the particle in a box. So it's a particle and infinitely deep potential that well has length L. And we know from boundary conditions that the wave function has to be zero at the boundaries. So when x is equal to zero and x is equal to L. And one simple function that satisfies that requirement is x times L minus x. So we see when x is zero, this term makes the function equal to zero. And when x is equal to L, this term makes the function equal to zero. So this would be a possible trial wave function that has some of the properties of the expected true wave function of the system. Another consideration that sometimes helps in choosing a trial wave function is that whatever our guess is, it should also be an eigenstate of any Hermitian operator which commutes with the Hamiltonian of this system we're interested in. What that means is that our guess or trial wave function cat phi should be an eigenstate of any Hermitian operator q that commutes with h, which means that they satisfy an eigen value equation. So if you operate with this Hermitian operator q on our trial wave function, it will give you an eigenvalue times that trial wave function. And usually when looking for operators that commute with the Hamiltonian, you typically use symmetry as a guide for this. So for example, if you know that your Hamiltonian is rotationally invariant, so it doesn't matter if you rotate your coordinate system, you get the same Hamiltonian, then you know that the angular momentum has to commute with the Hamiltonian. Finally, an important note that needs to be taking to account is even if we make a really good guess for our trial wave function, which means that our estimated energy is extremely close to the true ground state energy of the system. This trial wave function is unlikely to also give good estimates for other observables. So if you try to calculate the expectation value of the position or the expectation value of the momentum, it may not give you good results for these observables, even if it gave you a good estimate for the energy. So that's something to be careful with. So in the next video, we'll employ some of these tips and tricks in an example where we try to estimate the ground state energy of the hydrogen atom. And it's a good example because we know what the answer should be, so we know how close we actually get using some educated guess for the wave function.