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Following Susskind. Quantum Mechanics. Theoretical Minimum. Page 286.

Define velocity as the time derivative of the average position.

$$ v=\frac{d\left<X\right>}{dt}= \frac{d}{dt} \int \psi(x,t)^* x \psi dx(x,t) $$

An operator $L$ evolves according to the following formula where $H$ is the Hamiltonian.

Assuming $L$ has no explicit time dependence,

$$ \frac{d}{dt}\left<L\right>=\frac{i}{\hbar}\left<[H,L]\right> $$

For a free particle, $H=\frac{p^2 }{2m}$ therefore

$$ v=\frac{i}{2m\hbar}\left<[P^2,X]\right> $$

Note that $[P^2 , X]=P[P,X]+[P,X]P.$

Substitute $[P,X]=-i\hbar$.

Then $v=\frac{\left<P\right>}{m}$.

There is also a path integral approach.

Following Susskind. Quantum Mechanics. Theoretical Minimum. Page 286.

Define velocity as the time derivative of the average position.

$$ v=\frac{d\left<X\right>}{dt}= \frac{d}{dt} \int \psi(x,t)^* x \psi(x,t)dx $$

An operator $L$ evolves according to the following formula where $H$ is the Hamiltonian.

Assuming $L$ has no explicit time dependence,

$$ \frac{d}{dt}\left<L\right>=\frac{i}{\hbar}\left<[H,L]\right> $$

For a free particle, $H=\frac{p^2 }{2m}$ therefore

$$ v=\frac{i}{2m\hbar}\left<[P^2,X]\right> $$

Note that $[P^2 , X]=P[P,X]+[P,X]P.$

Substitute $[P,X]=-i\hbar$.

Then $v=\frac{\left<P\right>}{m}$.

There is also a path integral approach.

Following Susskind. Quantum Mechanics. Theoretical Minimum. Page 286.

Define velocity as the time derivative of the average position.

$v=\frac{d<X>}{dt}= \frac{d}{dt} \int \psi(x,t)^* x \psi dx(x,t)$

An operator $L$ evolves according to the following formula where $H$ is the Hamiltonian.

Assuming $L$ has no explicit time dependence,

$\frac{d}{dt}<L>=\frac{i}{\hbar}<[H,L]>$

For a free particle, $H=\frac{p^2 }{2m}$ therefore

$v=\frac{i}{2m\hbar}<[P^2,X]>$

Note that $[P^2 , X]=P[P,X]+[P,X]P.$

Substitute $[P,X]=-i\hbar$.

Then $v=\frac{<P>}{m}$.

There is also a path integral approach.

Following Susskind. Quantum Mechanics. Theoretical Minimum. Page 286.

Define velocity as the time derivative of the average position.

$$ v=\frac{d\left<X\right>}{dt}= \frac{d}{dt} \int \psi(x,t)^* x \psi dx(x,t) $$

An operator $L$ evolves according to the following formula where $H$ is the Hamiltonian.

Assuming $L$ has no explicit time dependence,

$$ \frac{d}{dt}\left<L\right>=\frac{i}{\hbar}\left<[H,L]\right> $$

For a free particle, $H=\frac{p^2 }{2m}$ therefore

$$ v=\frac{i}{2m\hbar}\left<[P^2,X]\right> $$

Note that $[P^2 , X]=P[P,X]+[P,X]P.$

Substitute $[P,X]=-i\hbar$.

Then $v=\frac{\left<P\right>}{m}$.

There is also a path integral approach.

Following Susskind. Quantum Mechanics. Theoretical Minimum. Page 286.

Define velocity as the time derivative of the average position.

$v=\frac{d<X>}{dt}= \frac{d}{dt} \int \psi(x,t)^* x \psi dx(x,t)$

An operator $L$ evolves according to the following formula where $H$ is the Hamiltonian.

$\frac{d}{dt}<L>=\frac{i}{\hbar}<[H,L]>$

For a free particle, $H=\frac{p^2 }{2m}$ therefore

$v=\frac{i}{2m\hbar}<[P^2,X]>$

Note that $[P^2 , X]=P[P,X]+[P,X]P.$

Substitute $[P,X]=-i\hbar$.

Then $v=\frac{<P>}{m}$.

There is also a path integral approach.

Following Susskind. Quantum Mechanics. Theoretical Minimum. Page 286.

Define velocity as the time derivative of the average position.

$v=\frac{d<X>}{dt}= \frac{d}{dt} \int \psi(x,t)^* x \psi dx(x,t)$

An operator $L$ evolves according to the following formula where $H$ is the Hamiltonian.

Assuming $L$ has no explicit time dependence,

$\frac{d}{dt}<L>=\frac{i}{\hbar}<[H,L]>$

For a free particle, $H=\frac{p^2 }{2m}$ therefore

$v=\frac{i}{2m\hbar}<[P^2,X]>$

Note that $[P^2 , X]=P[P,X]+[P,X]P.$

Substitute $[P,X]=-i\hbar$.

Then $v=\frac{<P>}{m}$.

There is also a path integral approach.