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I have some questions about linear algebra. Let's say $\{|v_1 \rangle, |v_2 \rangle, |v_3 \rangle \}$ are orthonormal basis of the $\mathcal{V}(\mathbb{C})$. Then, let's define two vectors $$ |a \rangle = 2i |v_1 \rangle - |v_ \rangle + 4 |v_3 \rangle \\ |b \rangle = |v_1 \rangle +3i |v_2 \rangle - |v_3 \rangle $$

My questions are:

1. Is it correct?

$$ \langle a|b \rangle = -2i \langle v_1|v_1 \rangle - 3i \langle v_2|v_2\rangle - 4 \langle v_3|v_3 \rangle = -2i - 3i -4 = -4 - 5i $$

or should there be some more expression above?

2. Does product of two kets $ |a\rangle|b\rangle$ make sens?

In my opinion no, because kets are columns and product of columns like that's is indefinite. But maybe I'm wrong and there is better explanation?

3. Does sum $ |\rangle + \langle| $ make sense?

Same as above. Bra is a row and ket is a column, so it's impossible to sum them.

Can somebody tell me if I'm wrong or explain 2. and 3. more mathematically?

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  • $\begingroup$ 1.When you turn a ket into a bra you have to conjugate the coefficients of the basis kets. $\endgroup$
    – bemjanim
    Feb 25, 2020 at 14:59

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For question $\mathbf{1}$ you're right: by using the orthonormality relation

$$\langle v_i | v_j \rangle= \delta_{ij} $$

you'll only get the scalar product of the same vectors since they give you zero with the others.

Question $\mathbf{2}$ is a little bit more complicated: the product $$|a\rangle|b\rangle$$ makes sense whenever the intended product is the tensor product. If we wanted to be precise we would make it clear by writing it as

$$ |a\rangle\otimes |b\rangle \equiv |a\rangle|b\rangle$$

This kind of product comes all the time, for example, when talking about system of particles. If you want to go deeper with an example search for addition of angular momentum on every QM book. For a deeper argument you can look up Fock space which is a fundamental concept in the quantum treatment of systems of many particles such as quantum field theories.

The last question it's easy: that sum makes no sense. Broadly speaking a ket is a vector in some Hilbert space while a bra is a functional acting on that Hilbert space. This means that a bra belongs to the dual of that Hilbert space. In this sense you clearly cannot add them since they do not belong to the same space to begin with.

Moreover I wanted to add that, while a it is a good way to think as bra and kets as row and column vectors, this is only one possible representation, the simples one. More abstractly a ket is an element of a certain Hilbert space with some properties and the bra is an element of the dual space. The two spaces are connected to one another thanks to Riesz representation theorem.

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  • $\begingroup$ Thank you. I understand what you said, but now I wonder about something similiar. I got why we can't sum bra and ket since they belong to the different space. But, why we can calculate the inner product of them like $ \langle a | b \rangle $ ? It doesn't matter here that's bra and ket don't belong to the same space? $\endgroup$
    – user237867
    Feb 25, 2020 at 16:02
  • $\begingroup$ @pepe The scalar product $(\quad,\quad)$ is inherited by the Hilbert space. With that you can define bras from their vectors in the Hilbert space: for every vector $|\psi\rangle\in H$ you can define a ket as $\langle\phi| = (\phi,\cdot)$. In this manner you can see that $\langle\phi|$ is a linear functional that can be applied on a ket as $$\langle\phi|(\psi) = (\phi, \psi) \equiv \langle\phi|\psi\rangle$$ All of this comes from the statement of Riesz representation theorem that tells us that every linear functional on an Hilbert space can be represented in a unique way with the inner product. $\endgroup$
    – Quiver
    Feb 25, 2020 at 17:21

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