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I am reading a derivation of the representation of the position operator $\hat{\mathbf{r}}$ in momentum space that goes like this:

$$ \langle\mathbf{p}|\hat{\mathbf{r}}|\psi\rangle =\int d^{3}x\langle\mathbf{p}|\mathbf{r}\rangle\langle\mathbf{r}|\hat{\mathbf{r}}|\psi\rangle =\int d^{3}xe^{-i\mathbf{p}\cdot\mathbf{r}/\hbar}\mathbf{r}\langle\mathbf{r}|\psi\rangle =\int\mathbf{r}\psi(\mathbf{r})e^{-i\mathbf{p}\cdot\mathbf{r}/\hbar}d^{3}x =i\hbar\nabla_{\mathbf{p}}\int\psi(\mathbf{r})e^{-i\mathbf{p}\cdot\mathbf{r}/\hbar}d^{3}x =i\hbar\nabla_{\mathbf{p}}\bar{\psi}(\mathbf{p}) $$

And I have trouble understanding the step $\langle\mathbf{r}|\hat{\mathbf{r}}|\psi\rangle=\mathbf{r}\langle\mathbf{r}|\psi\rangle$, since I'm used to thinking of the inner product on the left as a scalar, and the right-hand side seems to be a vector.

This similarly leads me to wonder how $\langle\mathbf{p}|\hat{\mathbf{r}}|\psi\rangle = i\hbar\nabla_{\mathbf{p}}\bar{\psi}(\mathbf{p})$ for the same reason, since the left seems to be a scalar and the right (due to $\nabla_{\mathbf{p}}$) is a vector.

Am I missing something simple?

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  • $\begingroup$ The analogy between vectors and brakets is the following: bras are row vectors, kets are column vectors and operators and matrices. In this case, $\hat{\mathbf{r}}$ is a "matrix vector", i.e. a collection of three matrices. Multiplying a bra, an operator and a ket will give a scalar. In this case, you are calculating the average of three operators ($\hat{x}$, $\hat{y}$, $\hat{z}$) which are usually represented as a vector (collection of three scalars). $\endgroup$ – TheAverageHijano Feb 22 at 0:55
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There are two different notions of vector at play here. The kets are vectors in the Hilbert space, and bra-kets are inner products in this vector space. The operator $\hat{\mathbf{r}}$ is a vector in a three-dimensional euclidean space. The matrix elements of such a vector operator are scalars with respect to the Hilbert space, but vectors in the euclidean space.

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