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Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription

$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $$

in the position basis. Now my question is, what if $x$ or $p$ appears in the denominator in a classical expression? How to promote this to a quantum expression? What would be the meaning of division by an operator?

Edit: Thank you for your responses. My expression likely contains a mixture of x and p. For eg., it could contain terms like $$\frac{p}{x^2}$$ or $$\frac{xp}{(x^2 + a^2)^{3/2}}$$. How to resolve products of non-commuting operators like x,p in a satisfactory way?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription

$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $$

in the position basis. Now my questions are:

• What if $x$ or $p$ appears in the denominator in a classical expression?

• How to promote this to a quantum expression? What would be the meaning of division by an operator?

My expression likely contains a mixture of $x$ and $p$. For e.g., it could contain terms like $$\frac{p}{x^2}$$ or $$\frac{xp}{(x^2 + a^2)^{3/2}}.$$

• How to resolve products of non-commuting operators like $x$, $p$ in a satisfactory way?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription

$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $$

in the position basis. Now my question is, what if $x$ or $p$ appears in the denominator in a classical expression? How to promote this to a quantum expression? What would be the meaning of division by an operator?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription

$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $$

in the position basis. Now my question is, what if $x$ or $p$ appears in the denominator in a classical expression? How to promote this to a quantum expression? What would be the meaning of division by an operator?

Edit: Thank you for your responses. My expression likely contains a mixture of x and p. For eg., it could contain terms like $$\frac{p}{x^2}$$ or $$\frac{xp}{(x^2 + a^2)^{3/2}}$$. How to resolve products of non-commuting operators like x,p in a satisfactory way?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription $$ X \rightarrow x, P \rightarrow i\hbar\frac{\partial}{\partial x} $$ in the position basis. Now my question is, what if x or p appears in the denominator in a classical expression. How to promote this to a quantum expression? What would be the meaning of division by an operator?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription

$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $$

in the position basis. Now my question is, what if $x$ or $p$ appears in the denominator in a classical expression? How to promote this to a quantum expression? What would be the meaning of division by an operator?