Part A Evaluate the average kinetic energie, Ek tic for the ground state n = 0 o

Question

Part A Evaluate the average kinetic energie, Ek tic for the ground state n = 0 of the harmonic oscillator by carrying out the appropriate integrations. Express your answer in terms of the variables , k, and appropriate constants. (Ekinetie- Submit My Answers Give Up Incorrect, One attempt remaining; Try Again Part B Evaluate the average potential energie, potential Express your answer in terms of the variables for the ground state n = 0 of the harmonic oscillator by carrying out the appropriate integrations. k, and appropriate constants. 0? potential Submit My Answers Give Up

Explanation / Answer

You can evaluate the average kinetic energy of the harmonic oscillator using the harmonic oscillator wavefunctions (see link below).

T = *(–²/2m)(d²/dx²)dx (– x ), where (–²/2m)(d²/dx²) is the kinetic energy operator.

The wavefunctions are given in textbooks and in the Wikipedia link below. You can do (look up) those integrals; they're straightforward if a little tedious. An easier way to do it is to use the Virial Theorem, which allows you to say that if the potential energy between particles has a power law form: V = x ( = constant), the average kinetic energy, T, and the average potential energy, V, are related by 2T = nV (see link below). For the Harmonic oscillator, V = ½kx², so T = V. For a quantum mechanical harmonic oscillator we know that the total energy is given by


E = (n + ½), where n = 0, 1, 2, ...
<T> = avg kinetic energy

<V> = avg. potential energy

E = T + V = 2T by the Virial Theorem

T = V = ½(n + ½)

for n=0 avg. kinetic energy

<T> = 1/4*

for n= 0 avg. potential energy

<V> = 1/4*

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