Zero-Point Energy: Calculating the Minimum Value

['AQF]

"Consider a particle with mass m moving in a potential U=1/2kx^2, as in a mass-spring system. The total energy of the particle is E=p^2/(2m)+1/2kx^2. Assume that p and x are approximately related by the Heisenburg Uncertainty Principal, px approximately equals h.
a) Calculate the minimum possible value of the energy E, and the value of x that gives this this minimum E. This lowest possible energy, which is not zero, is called the zero-point energy.
b) For the x calculated in part (a), what is the ratio of kinetic to potential energy of the particle?" -University Physics, by Young and Freedman pg. 1517

I do not know how to answer this question.
For (a), I assume that you need to take the derivative of E=p^2/(2m)+1/2kx^2 to minimize it, but in respect to what variable?
For (b), since E=U+KE, the ratio must be (E-1/2kx^2)/(1/2kx^2), but I am unsure of what that would be without part (a).

Thanks for your help!

 

[Physics Monkey]

The problem tells you to assume that p and x are approximately related in a certain specific way by the Heisenberg Uncertainty Principle. Try solving this relation for p in terms in x and plugging into the Hamiltonian. Can you find a minimum of the resulting expression?

This is a standard way to estimate the ground state energy of a bound system.

 

['AQF]

What is the "Hamiltonian" that you refer to?
Thanks for your help.

 

[Physics Monkey]

Sorry, the Hamiltonian is just the energy.

 

['AQF]

What energy (E, KE, or U)?

 

[Physics Monkey]

Come on now, AQF, work with me here. I can't just tell you answer. What energy are you trying to minimize?