Do bras and kets have dimensions?

Question

I'm trying to understand more intuitively what bras and kets are, but some aspects of them remain a mystery to me.

We usually think of $\psi (x)$ as having dimension of $[1/\sqrt{L}]$ so that squaring it and multiplying it with a distance differential would result in a dimensionless quantity. An example of this is:

$$\int_{-\infty}^{\infty} \mid\psi(x)\mid^2 dx= 1$$ for normalized wavefunctions. I also know it is possible to write a ket in position basis as:

$$\mid \psi \rangle = \int_{-\infty}^{\infty} \psi(x) \mid x \rangle dx$$

I would like to believe that $ \mid \psi \rangle$ has no units, it can be represented in the position or momentum base, so it having units doesn't make a whole lot of sense, but this leads me to the conclusion that $\mid x \rangle$ must have the same units as the wavefunction in order to cancel the lenght units of $dx$!

Is this correct? If so, what is its physical interpretation? do position and momentum kets have units after all?

Answer 1

This is a very interesting question. I don't know if there is a general and definitive answer, but I'll try to make some comments. I apologize if this ends up rambling; I'm finding this out as I write this answer.

Operators have dimensions, since their eigenvalues are physical quantities. For bras and kets it gets more complicated. First, you cannot in general say that they are dimensionless. To see why, consider a state with a certain position $|x\rangle$. Since $\langle x | x' \rangle = \delta(x-x')$ and the Dirac delta has the inverse dimension of its argument, it must be that $[ \langle x | ] \times [ | x \rangle ] = 1/L$. A similar relationship holds for momentum eigenstates. Of course, there are higher powers of $L$ in higher dimensions.

However, consider an operator with discrete spectrum, such as the energy in an atom or something like that. Then the appropriate equation is $\langle m | n \rangle = \delta_{mn}$, and since this delta is dimensionless, bras and kets must have inverse dimensions. This gets even weirder when you consider that the Hamiltonian for a hydrogen atom has both discrete and continuous eigenvalues, so the relationship between the bras and the kets' dimensions will be different depending on the energy (or whatever physical quantity is appropiate).

We have the equation $\langle x | p \rangle = \frac1{\sqrt{2\pi\hbar}} \exp(ipx/\hbar)$. I at first thought that this combined with $[\langle x |] \times [| p \rangle ] = [\langle p |] \times [| x \rangle ]$ would allow us to find the dimensions of $|x\rangle$ (and everything else), but it turns out that the normalization conditions of $|x\rangle$ and $|p\rangle$ force the dimensions of $\langle x | p \rangle$ to come out right. We can find that $[|p\rangle] = \sqrt{T/M} [|x \rangle]$, but we can't go any further. Similar relationships will apply for the eigenstates of your favourite operator.

Any given ket is a linear combination of eigenkets, but again there are subtleties depending on whether the spectrum is discrete or continuous. Suppose we have two observables $O_1$ and $O_2$ with discrete spectrum and eigenstates $|n\rangle_1$ and $|n\rangle_2$. Any state $|\psi\rangle$ can be expressed as a dimensionless linear combination of the eigenstates (dimensionless because since $\langle n | n \rangle = 1$, the squares of the coefficientes make up probabilities): $|\psi\rangle = \sum_n a_n |n\rangle_1 = \sum_n b_n |n\rangle_2$. This implies that the eigenkets of all observables with discrete spectrum have the same dimensions, and likewise for the eigenbras.

It gets trickier for observables with continuous spectrum such as $x$ and $p$, because of the integration measure. We have $|\psi\rangle = \int f(x) |x\rangle\ dx = \int g(p) |p\rangle\ dp$. $\langle \psi | \psi \rangle = 1$ implies $\int |f(x)|^2\ dx = 1$, so that $[f] = 1/\sqrt{L}$ and likewise $[g] = \sqrt{T/ML}$. This should be no surprise since $f$ and $g$ are Fourier transforms of each other, with an $1/\sqrt{\hbar}$ thrown in. From this we can deduce $[|p\rangle] = \sqrt{T/M} [|x \rangle]$, which we already knew, and $\sqrt{L} [|x \rangle] = [|n \rangle]$.

The conclusion seems to be the following. All eigenkets with discrete eigenvalues must have the same dimensions, but it looks like that dimension is arbitrary (so you could take them to be dimensionless). Furthermore, normalized states have that same dimension. Eigenstates with continuous spectrum are more complicated; if we have an observable $A$ (with continuous eigenvalues) with eigenvalues $a$, then we can use the fact that $|\psi\rangle$ can be written either as an integral over eigenstates of $A$ or as a sum over discrete eigenstates to find that $\sqrt{[a]} [|a\rangle] = [|n\rangle]$, where $|n\rangle$ is some discrete eigenket. So once you fix the dimensions of one ket, you fix the dimensions of every other ket.

Answer 2

This question is fun, and I'm not sure there is really a right answer, so I wanted to offer a slightly different perspective.

What are units all about? Units tell us how quantities transform under a change of scale. The reason we say a distance $d$ has units of length, is that the numerical value assigned to $d$ will be rescaled by a factor of 10 if we decide to use millimeters rather than centimeters.

On the other hand, we have quantum mechanics, where states are represented by rays in Hilbert space. In other words, we consider two states, $|\chi\rangle$ and $|\phi\rangle$, to be physically equivalent if they are related by $|\chi\rangle = \lambda |\phi \rangle$ for some complex number $\lambda$.

So the question is, if we change our unit of length, by what factor should we rescale our state vector?

Well, typically we deal with the equivalence of different states in a ray by choosing to normalize our states in some way. So the question is, should the normalization of the states change under a rescaling of our reference for length? (or time or mass or ...)

For a discrete set of eigenstates labeled by an integer $|n\rangle$, we can sensibly choose to normalize the states like $\langle m | n \rangle = \delta_{m,n}$, where the right hand side is dimensionless. Then it doesn't make sense for the normalization condition to scale with length.

For continuous eigenstates, the states are not truly normalizable, but only normalizable by a delta function $\langle x | y \rangle = \delta(x-y)$. In order to avoid introducing an arbitrary constant with units of length on the right hand side, it makes more sense for the normalization of our states to scale with dimensions of $L^{-1/2}$.

One way to interpret the distinction between continuous and discrete eigentstates, is to note the following. $|\langle n | \psi \rangle|^2$ is the probability to observe the state to have eigenvalue $n$, while $|\langle x | \psi \rangle|^2$ is a probability density for a particle to be located at $x$.

Answer 3

This question is investigated in an article from 2020 'Do bras and kets have dimensions?'. In brief, the units of a bra or ket have some freedom, as mentioned by some of the other answerers. A useful convention is one where the units of a bra and ket are equal. If we have normalised states where $$\langle\alpha\vert\alpha\rangle=1,$$ then the both the bra and ket will be dimensionless. Their results are summarised in Table 1 from the manuscript:

If we have a continuous basis however: $$\langle x\vert y\rangle=\delta(x-y),$$ then the units of the delta function imply that the bras and kets will have units. Repeating the arguments in the paper, we may again take these to be the same. In the position basis, this gives both the bra and ket units of $1/\sqrt{L}$.

Answer 4

Really good question.

Measurements have unit but in quantum mechanics, a measurement is the "evaluation" of an observable on a state (or a state on an observable) something like $$ \left\langle \psi | A | \psi \right\rangle ,\quad \psi\in\mathcal{H},\ A\in\mathcal{B}(\mathcal{H}) \ \text{self-adjoint}$$ A priori, there seem to be an arbitrariness in the choice of unit for $|\psi\rangle$ and $A$ but usually the measurement is a convex combination of the eigenvalues of $A$ (or an integral of a density of probability over the spectrum) so that the eigenvalues really have unit that of the observable (length it the observable is the position, Mass times speed if it is the momentum observable)

If $\left\lVert \psi\right\rVert^2 $ has interpretation of density of probability of presence it has unit $\frac{1}{Volume}$ (or length in 1 dimension) which cancels the $dx$ of integration (as you have written in your first equation). So if $|\psi\rangle$ has no dimension, then $|x\rangle$ indeed has dimension $\frac{1}{\sqrt{L}}$ (L length).

OOOhhhh, big possible confusion coming from quantum field theory (and also the case in second quantization of schrödinger eq. ) when one write the action as a functional of the fields, then these have dimensions, but they are operators and play a role that is more similar to the observable $A$ rather than a state.