Harmonic Oscillator: Position Expectation Value & Ground State

[matrix5]

why is the expectation value of the position of a harmonic oscillator in its ground state zero? and what does it mean that it is in ground state? is ground state equal to n=0 or n=1?

 

[DaTario]

Ground state of a SHO has a gaussian wave function in which the average position is calculated in a standard way yielding x_avg = 0 as result (of course the center of the gaussian is suposed to be at x = 0 too).

The energy can be calculated with the n quantum number, so that n = 0 corresponds to the energy of 1/2 \hbar \omega. Answering your sencond question, n=0 corresponds to groud state.

 

[R0man]

The expectation value of the position of a harmonic oscillator in its ground state is zero because the ground state of a harmonic oscillator is defined as the state with the lowest energy. This means that the particle in the oscillator is in its most stable and balanced state, with no net movement or displacement from its equilibrium position.

In quantum mechanics, the ground state is represented by the quantum number n=0, indicating that the oscillator is in its lowest energy state and has no excess energy to "jump" to a higher energy level. This is in contrast to the first excited state, represented by n=1, where the oscillator has slightly more energy and is displaced from its equilibrium position.

Therefore, the ground state is the lowest energy state of the harmonic oscillator and corresponds to the expectation value of the position being zero. This means that in the ground state, the particle in the oscillator is most likely to be found at its equilibrium position.

In summary, the ground state of a harmonic oscillator is the state with the lowest energy and the expectation value of the position is zero because the particle is in its most stable and balanced state at its equilibrium position. The ground state is equivalent to n=0 in quantum mechanics.