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Qmechanic ♦
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Expectation values of x$x$, y$y$, z$z$ in hydrogen?

The expectation value of $r=\sqrt{x^2 + y^2 + z^2}$ for the electron in the ground state in hydrogen is $\frac{3a}{2}$ where a is the bohr radius.

I can easily see from the integration that the expectation values of $x$ , $y$, $z$ individually is $0$ because of the factor $\cos\phi$, $\sin\phi$ and $\cos\theta$(.$\sin\theta$) respectively for $x$ $y$, $z$ integrating out to $0$.

What does this mean? Is it an artifact of the spherically symmetric potential? Since $r$ has a non zero value but what makes up $r$ ie $x$, $y$, $z$ (or rather the squareroot of the sum of their squares) each has individually $0$ and i can simultaneously determine their values since they commute with each other, which means $x,y,z$ are all simultaneously $0$ and $r$ is not  ?!

Expectation values of x, y, z in hydrogen?

The expectation value of $r=\sqrt{x^2 + y^2 + z^2}$ for the electron in the ground state in hydrogen is $\frac{3a}{2}$ where a is the bohr radius.

I can easily see from the integration that the expectation values of $x$ , $y$, $z$ individually is $0$ because of the factor $\cos\phi$, $\sin\phi$ and $\cos\theta$(.$\sin\theta$) respectively for $x$ $y$, $z$ integrating out to $0$.

What does this mean? Is it an artifact of the spherically symmetric potential? Since $r$ has a non zero value but what makes up $r$ ie $x$, $y$, $z$ (or rather the squareroot of the sum of their squares) each has individually $0$ and i can simultaneously determine their values since they commute with each other, which means $x,y,z$ are all simultaneously $0$ and $r$ is not  ?!

Expectation values of $x$, $y$, $z$ in hydrogen?

The expectation value of $r=\sqrt{x^2 + y^2 + z^2}$ for the electron in the ground state in hydrogen is $\frac{3a}{2}$ where a is the bohr radius.

I can easily see from the integration that the expectation values of $x$ , $y$, $z$ individually is $0$ because of the factor $\cos\phi$, $\sin\phi$ and $\cos\theta$(.$\sin\theta$) respectively for $x$ $y$, $z$ integrating out to $0$.

What does this mean? Is it an artifact of the spherically symmetric potential? Since $r$ has a non zero value but what makes up $r$ ie $x$, $y$, $z$ (or rather the squareroot of the sum of their squares) each has individually $0$ and i can simultaneously determine their values since they commute with each other, which means $x,y,z$ are all simultaneously $0$ and $r$ is not?!

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Prasad Mani
Prasad Mani
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Expectation values of x, y, z in hydrogen?

The expectation value of $r=\sqrt{x^2 + y^2 + z^2}$ for the electron in the ground state in hydrogen is $\frac{3a}{2}$ where a is the bohr radius.

I can easily see from the integration that the expectation values of $x$ , $y$, $z$ individually is $0$ because of the factor $\cos\phi$, $\sin\phi$ and $\cos\theta$(.$\sin\theta$) respectively for $x$ $y$, $z$ integrating out to $0$.

What does this mean? Is it an artifact of the spherically symmetric potential? Since $r$ has a non zero value but what makes up $r$ ie $x$, $y$, $z$ (or rather the squareroot of the sum of their squares) each has individually $0$ and i can simultaneously determine their values since they commute with each other, which means $x,y,z$ are all simultaneously $0$ and $r$ is not ?!