The mean value in Quantum Mechanics is the average result obtained when a large number of measurements are made on identical systems, i.e. systems in the same state |y(t)>.

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The mean value of X and P in an eigenstate of H is always zero.

The classical mean value can be defined as the average result obtained when a large number of measurements are made on the same system at different times, or when a large number of measurements are made at the same time on different systems, whose phases are randomly chosen.

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Consider a harmonic oscillator of well-defined energy.

For a harmonic oscillator of well-defined energy .

But we have to remember that the classical and quantum mechanical mean value are interpreted differently.

In a similar way the quantum mechanical and the classical root-mean-square deviation are interpreted differently.

In Quantum Mechanics we define

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for n=0, i.e. for the ground state.

For a classical harmonic oscillator with energy we have

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The quantum mechanical root mean square deviations are proportional to the classical maximum amplitude and maximum momentum.

The classical root-mean-square deviation is given by

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For a harmonic oscillator of well defined energy we obtain the same root-mean-square deviations.  But again we have to remember that the classical and quantum mechanical root-mean-square deviation are interpreted differently.

Coherent states

Ehrenfest’s theorem states .

In an eigenstate of the Hamiltonian <X>(t) and <P>(t) are zero at all times.  However if the state vector |y(0)> is a linear superposition of energy eigenstates then <X>(t) and <P>(t) satisfy the classical equations of motion and behave like the classical position and momentum as a function of time.

Proof:

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                            a+bi             a-bi

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where

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Similarly,

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Conclusion

If we want a quantum mechanical wave packet to behave like a classical particle we have to build it as a linear superposition of energy eigenstates.

Link:

Problem:

For a one-dimensional simple harmonic oscillator we may define raising and lowering operators

with properties

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(a) Show by direct calculation that the ground state of the oscillator satisfies
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and hence is a minimum uncertainty state.

(b) A coherent state of a one-dimensional oscillator, |a>, may be obtained by applying the finite displacement operator to the ground state eigenket |y(0)>.  Here a is the displacement distance.  Using this definition of a coherent state, prove that the coherent state is also a minimum uncertainty state.