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On my introductory course in Quantum Mechanics, the uncertainty of an operator $A$ in the state $\psi$ is defined by

$$(\Delta A)^2_{\psi}=\langle(A-\langle A \rangle_{\psi})^2\rangle _{\psi}$$

I'm having trouble extracting meaning from the R.H.S. If $A$ is a self-adjoint operator then if the underlying hilbert space has finite dimension, $A$ is a matrix. However, $\langle A\rangle_{\psi}$ is a number. How could one be subtracted from the other.

I think my lack of understanding stems from my lack of knowledge about operators. For example, I know that there is something called the momentum operator $p$, but to my knowledge it is not a matrix; $p=-i\hbar\frac{\partial}{\partial x}$.

Can anyone clarify?

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1 Answer 1

You are right that $\langle A\rangle_{\psi}$ is a number and not an operator. However, people often write just a number when they actually mean the identity operator times that number. So in the right hand side, $\langle A\rangle_{\psi}$ should actually be $\langle A\rangle_{\psi} \mathbf{1}_H$, where $\mathbf{1}_H$ is the identity operator on your hilbert space.

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Oh ok I see, and is a scalar times the identity operator defined? I'm guessing for example $2\mathbf{1}_H \psi=2\psi$? –  James Machin Mar 31 at 20:33
    
@JamesMachin Essentially a scalar times an operator is a scalar times a matrix, which is perfectly well defined. And the identity operator just corresponds to the identity matrix, i.e. a diagonal matrix with all diagonal elements equal to $1$. So what you write there is correct. –  Wouter Mar 31 at 20:52

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