In this problem, you will derive the ground-state energy
Chapter , Problem 37(choose chapter or problem)
In this problem, you will derive the ground-state energy of the harmonic oscillator using the precise form of the uncertainty principle, where and are defined to be the standard deviations and Proceed as follows: 1. Write the total classical energy in terms of the position and momentum using and 2. Show that and Hint: See Equations 17-34a and 17-34b. 3. Use the symmetry of the potential energy function to argue that and must be zero, so that and 4. Assume that to eliminate from the average energy and write as where 5. Set to find the value of for which is a minimum. 6. Show that the minimum energy is given by
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