Background: In quantum mechanics, one usually starts by taking the classical Hamiltonian and creating a Hamiltonian operator. One then calculates the energy spectrum for the system by solving the time independent Schrodinger equation. This is almost always a lot of work. Sometimes, one is interested in the minimum energy that a system can achieve, sometimes referred to as the "zero point energy". As we know from previous problems, the Heisenberg uncertainty principle prevents the ground state energy of a particle system to reach zero energy. Given the following Hamiltonian: E =+ Ax*. a) Estimate the ground state energy. Hint: Assume you can replace (x*) with (x²)². Use the Heisenberg uncertainty principle as a constraint and then minimize the energy funetion. b) Do you expect (x*)* to be larger or smaller than (x*). Explain using mathematical justification.

Background: In quantum mechanics, one usually starts by taking the classical Hamiltonian and creating a Hamiltonian operator. One then calculates the energy spectrum for the system by solving the time independent Schrodinger equation. This is almost always a lot of work. Sometimes, one is interested in the minimum energy that a system can achieve, sometimes referred to as the "zero point energy". As we know from previous problems, the Heisenberg uncertainty principle prevents the ground state energy of a particle system to reach zero energy. Given the following Hamiltonian: E =+ Ax. a) Estimate the ground state energy. Hint: Assume you can replace (x) with (x²)². Use the Heisenberg uncertainty principle as a constraint and then minimize the energy funetion. b) Do you expect (x) to be larger or smaller than (x*). Explain using mathematical justification.