- #1

- 93

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## Main Question or Discussion Point

Hi,

Bohm discusses several ambiguities is the hermitization of operators that consist of products op p's and x's (page 184 and further)

He reaches the conclusion that an operator [itex]p^m x^n[/itex] should be replaced by

[tex]

\frac{p^m x^n + x^n p^m}{2}

[/tex]

on the grounds that this way the expectation for the operator is always real. But this method then regard the DIFFERENT operators like p^2x, pxp and xp^2 as being the same.

This Hermitization seems to me a very shakey procedure with little ground in actual physics, or is there more to it?

Bohm goes on the describe another ambiguity. For the operator (px)^2, should we use

[tex]

\frac{p^2 x^2 + x^2 p^2}{2}

[/tex]

or

[tex]

\left( \frac{xp+px}{2} \right)^2

[/tex]

He than poses the problem to show that the above are not the same but differ by quantities of order [itex]\hbar^2[/itex].

I do obtain a difference of [itex]3\hbar^2/4[/itex] But then Bohm claims that this difference can be neglected and the two alternatives for expressing the operator seen als FAP equivalent because the difference is so small.

But The ENTIRE operators are of the order [itex]\hbar^2[/itex] (as can easily be seen since each term contains p twice) so I don't see how his argument holds.

Lastly, Bohm suggests that in the future experimental results might differentate between the two ways of expressing the operator. Well, this is the future, so is there any news on that?

Thanks,

A_B

Bohm discusses several ambiguities is the hermitization of operators that consist of products op p's and x's (page 184 and further)

He reaches the conclusion that an operator [itex]p^m x^n[/itex] should be replaced by

[tex]

\frac{p^m x^n + x^n p^m}{2}

[/tex]

on the grounds that this way the expectation for the operator is always real. But this method then regard the DIFFERENT operators like p^2x, pxp and xp^2 as being the same.

This Hermitization seems to me a very shakey procedure with little ground in actual physics, or is there more to it?

Bohm goes on the describe another ambiguity. For the operator (px)^2, should we use

[tex]

\frac{p^2 x^2 + x^2 p^2}{2}

[/tex]

or

[tex]

\left( \frac{xp+px}{2} \right)^2

[/tex]

He than poses the problem to show that the above are not the same but differ by quantities of order [itex]\hbar^2[/itex].

I do obtain a difference of [itex]3\hbar^2/4[/itex] But then Bohm claims that this difference can be neglected and the two alternatives for expressing the operator seen als FAP equivalent because the difference is so small.

But The ENTIRE operators are of the order [itex]\hbar^2[/itex] (as can easily be seen since each term contains p twice) so I don't see how his argument holds.

Lastly, Bohm suggests that in the future experimental results might differentate between the two ways of expressing the operator. Well, this is the future, so is there any news on that?

Thanks,

A_B