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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 ground state energy of a quantum system. There is a quick method that allows one to estimate 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*. 2m Estimate the ground state energy. Hint: Assume you can replace (x*) with (x?)². Use the a) Heisenberg uncertainty principle as a constraint and then minimize the energy: function.

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 ground state energy of a quantum system. There is a quick method that allows one to estimate 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*. 2m Estimate the ground state energy. Hint: Assume you can replace (x*) with (x?)². Use the a) Heisenberg uncertainty principle as a constraint and then minimize the energy: function.

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**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 Schrödinger equation. This is almost always a lot of work. Sometimes, one is interested in the ground state energy of a quantum system. There is a quick method that allows one to estimate 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: \( E = \frac{p^2}{2m} + Ax^4 \). a) **Estimate the ground state energy. Hint:** Assume you can replace \(\langle x^4 \rangle\) with \((\langle x^2 \rangle)^2\). Use the Heisenberg uncertainty principle as a constraint and then minimize the energy function.
Transcribed Image Text:**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 Schrödinger equation. This is almost always a lot of work. Sometimes, one is interested in the ground state energy of a quantum system. There is a quick method that allows one to estimate 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: \( E = \frac{p^2}{2m} + Ax^4 \). a) **Estimate the ground state energy. Hint:** Assume you can replace \(\langle x^4 \rangle\) with \((\langle x^2 \rangle)^2\). Use the Heisenberg uncertainty principle as a constraint and then minimize the energy function.
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