Estimating Energy using Uncertainty Principle

Question:

Estimate the ground state energy of a particle of mass m and potential energy V(x) = ax⁶ using the uncertainty principle.

Solution:
Energy of the particle is given by:

For this one dimensional system, Heisenberg uncertainty principle is given by:

and, at ground state (with minimum uncertainty), it can be written as:

or:

By substituting p in terms of x in equation (1), we can find:

E has minimum value when its derivative with respect to x is zero, and its second derivative is positive.

So, the value of x when energy is minimum, can be found as:

Now, by substituting this value of x in equation (5), we can find ground state energy:

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