What does $\lvert \psi \rangle$ mean?

Question

What does the notation $|\psi\rangle$ mean and how does it relate to the notation $|0\rangle$ and $|1\rangle$? Does $\psi$ have some connection to the one found in Shroedinger's equation?

Answer 1

The notation $\lvert \alpha \rangle$, $\lvert \psi \rangle$, etc. just indicates that the thing in question is a vector. Furthermore, it is extremely common that the following things are also intended (in that one really should say if any of these things do not hold):

• $\lvert \psi \rangle$ is a column vector.
• $\lvert \psi \rangle$ is not the zero vector.
• $\lvert \psi \rangle$ has complex coefficients (and in particular its not necessarily restricted to real coefficients).
• $\lvert \psi \rangle$ has unit Euclidean norm.

Such a vector is usually called a state-vector, or sometimes a (pure) state.

The symbol $\lvert \psi \rangle$ in particular, like a bold-face $\mathbf v$ in the context of vector algebra, is often used just as a generic choice of symbol or variable. It does relate to the wave function $\psi$ in Schrödinger's equation, in that the wave function $\psi$ can be interpreted as a state vector in a very large vector space.

The notation of putting $\psi$ in asymmetrical brackets (technically, a "ket": one reads $\lvert \psi \rangle$ either as "psi" or "ket-psi" by convention) is a part of Dirac notation. There is a corresponding dual object $\langle \psi \rvert = \lvert \psi \rangle^\dagger$ called a "bra" (we read $\langle \psi \rvert$ as "bra-psi"), and when you multiply a bra and a ket, you evaluate an inner product (or a "bracket"): $$ \bigl( \langle \alpha \rvert \bigr) \bigl( \lvert \psi \rangle \bigr) = \langle \alpha \vert \psi \rangle = \langle \alpha, \psi \rangle.$$ Thus, you'd often see the normalisation condition on state vectors written as $\langle \psi \vert \psi \rangle = 1$. You will often see people talk about some state vectors $\lvert \alpha \rangle$ and $\lvert \beta \rangle$ being orthogonal: this can be denoted by $\langle \alpha \vert \beta \rangle = 0$.

The symbols $\lvert 0 \rangle$ and $\lvert 1 \rangle$ are a (very frequently occurring) special case. They indicate standard basis vectors, usually of $\mathbb C^2$, which have a 1 in one position and 0 everywhere else, and which are orthogonal to one another: $$ \lvert 0 \rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad\lvert 1 \rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. $$ Elsewhere in mathematics, these might be written instead as $\mathbf e_0$ and $\mathbf e_1$.

• It's reasonably common to consider such vectors indexed by other integers, e.g. $\lvert 2 \rangle$, $\lvert 3 \rangle$, etc.: these are then standard basis vectors $\mathbf e_2$, $\mathbf e_3$, etc. in a vector space of dimension greater than 2.

• It is also common to see vectors indexed by binary strings, e.g. $\lvert 001 \rangle$: this is a standard basis state of a system consisting of multiple qubits, in this case identified with the Kronecker product $\lvert 0 \rangle \otimes \lvert 0 \rangle \otimes \lvert 1 \rangle = \mathbf e_0 \otimes \mathbf e_0 \otimes \mathbf e_1 =: \mathbf e_{001}$.

• It is also common to see vectors indexed by elements of an arbitrary (though usually finite) set, e.g. $\lvert v \rangle$ for a vertex $v$ of some graph $G$. This is just a standard basis vector in some vector space, indexed by (the elements of) the set in question.

• More generally, it is very very common to write $\lvert \text{(expression)} \rangle$, where $\text{(expression)}$ is some variable or formula on variables ranging over some set. This just represents a standard basis vector which is being indexed by that expression. For instance, to take an arbitrary example, for $x,y, z\in \mathbb Z_8$, we may write $\lvert z{+} xy \rangle =: \mathbf e_{z{+}xy} $ (a standard basis state of a vector space of dimension eight, indexed by elements of $\mathbb Z_8$), in which case we should understand from context that "$+$" is the addition operation in the ring of the integers modulo 8, and "$xy$" is the product of $x$ and $y$ in that same number system.

There are more special cases in Dirac notation, which might not be the most sensible notation from a general point of view (they are the equivalent of "irregular verbs") but which are in heavy use and which are useful shorthand:

• $\lvert + \rangle$ represents the specific state $\tfrac1{\sqrt 2}(\lvert 0 \rangle + \lvert 1 \rangle)$.

• $\lvert - \rangle$ represents the specific state $\tfrac1{\sqrt 2}(\lvert 0 \rangle - \lvert 1 \rangle)$.

• Generalising the correspondence above with the symbols $+$ or $-$ in the kets, and the idea that you can have expressions in your kets, the symbol $\lvert \pm \rangle$ represents either one (or the set of both) of the two state vectors $\tfrac1{\sqrt 2}(\lvert 0 \rangle \pm \lvert 1 \rangle)$.

• Physicists often put arrows in their kets: for example, $\lvert \uparrow \rangle$, $\lvert \downarrow \rangle$, $\lvert \rightarrow \rangle$, $\lvert \nearrow \rangle$, and others generally refer to spin states or polarisations in physical systems, where the arrow literally refers to some spacial orientation.

In general, you can put pretty much any symbol in a bra or a ket that you like, and it will be immediately understood to refer to some state or functional. Because of the conventions above, this can allow you to quickly import some meaning which the symbols might have in a closely related context. (Though you should choose your notation carefully and explain it if it is not in very common usage.)

Hopefully this clarifies some of what's going on with the notation.