Question

2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy behavior of quantu systems. It is used constantly in marny-body physics and in quantum chemistry. The main idea is thst for any physical Hamiltonian, there is a lowest energy state, i.e. the ground state Ipo). All other states (ignoring degeneracy) have higher energy that this one. Therefore we have Therefore, to get an upper bound on the energy of Eo, it suffices to pick a good trial wave function and simply compute the expected energy. The better your choice of trial wave function, the closer you'll get to the true ground state energy Often, we exploit our physical intuition about the Hamiltonian to design a family of normalized trial wave functions that have one or more variational parameters that can be optimized over to get the best energy estimate among the wave functions in that family. That is, given a set a-fa,....an of parameters and a family of trial wave functions (a)), we find For example, we can consider the following Gsussian trial wave function in position spece and minimize the energy of a given Hamiltonian over the parameters A and b. Apply the Gaussian trial wave function to find the smallest expected energy you can for the following potentials written in the position basis. Be sure to include a momentum term in your so that (s) The linear potentil, V()-ar (b) The quartic potentil,V() az* Now consider a potential with a cusp, given by V() a(r-r). It will be convenient to rescale r and p such that the total Hamiltonian for this potential is given by c)Consider a trial wave function that is a positive superposition of tuo Gaussian trial wave functions centered at differentヱcoordinates Compute numerically if necessary, your best estimate of the ground state energy. Plot your answer in units of w as a function of the separation radius r (d) Comment on your answer in the two limiting cases of r1 and r1. Compare esch of your answers in these limits to another system that you already understand. Do your answers agree? The remainder of this question is OPTIONAL and will not be assessed. It is merely here for your edification The first excited state can also be studied using the variational method. To do this, we first find a trial ground state wave funetion l')(a) with some set of optimized variational parameters a*, so that our estimate for the ground state energy isWolHa)- Then we pick a new variational family of normalized wave functions (b)) with new variational paraneters b. The energy for the first excited state is then approximated by minimizing over all trial wave functions that are orthogonal to the trial ground state,

Answer 1

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