The uncertainty principle is a fundamental limit to the precision with which we can measure certain pairs of observables, such as position and momentum. See the tips for more background on the uncertainty principle.

Instead of using the wavefunction to calculate this expectation value directly, we can use the energy of the wavefunction to simplify the calculations needed. The energy of the ground state of the harmonic oscillator is given below.

We know that the wavefunction is even. The square of an even function is even as well, so we can pull out a factor of 2 and change the lower bound to 0.

The uncertainty of an observable such as position is mathematically the standard deviation. That is, we find the average value, take each value and subtract from the average, square those values and average, and then take the square root.

This differential equation has variable coefficients and cannot easily be solved by elementary methods. However, after normalizing, the solution for the ground state can be written like so. Remember that this solution only describes a one-dimensional oscillator.

We see that the Hamiltonian does not depend explicitly on time, so the solutions to the equation will be stationary states. The time-independent Schrödinger equation is an eigenvalue equation, so solving it means that we are finding the energy eigenvalues and their corresponding eigenfunctions - the wavefunctions.

While the position and momentum variables have been replaced with their corresponding operators, the expression still resembles the kinetic and potential energies of a classical harmonic oscillator. Since we are working in physical space, the position operator is given bywhile the momentum operator is given by

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