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