The Schrödinger Equation in Dirac Notation

The physical state of a system is represented by a set of probability amplitudes (wave functions), which form a linear vector space. This linear vector space is a particular type of space called a Hilbert Space. Another way to think about the Hilbert space is as an infinite dimensional space of square normalizable functions. This is analogous to 3-dimensional space, where the basis is $\left({\hat {i}},{\hat {j}},{\hat {k}}\right)$ in a generalized coordinate system. In the Hilbert space, the basis is formed by an infinite set of complex functions. The basis for a Hilbert space is written like $\left(|\psi _{0}\rangle ,|\psi _{1}\rangle ,|\psi _{2}\rangle ,...,|\psi _{j}\rangle ,...\right)$, where each $|\psi _{i}\rangle$ is a complex vector function.

We denote a state vector $\psi \$ in Hilbert space with Dirac notation as a “ket” $|\psi \rangle$, and its complex conjugate (or dual vector) $\psi \$* is denoted by a “bra” $\langle \psi |$.

Therefore, in the space of wavefunctions that belong to the Hilbert space, any wavefunction can be written as a linear combination of the basis function: $|\phi \rangle =\sum _{n}c_{n}|\psi _{n}\rangle$, where $c_{n}$ denotes a complex number.

By projecting the state vector $|\psi \rangle$ onto different basis, we can obtain the wavefunctions of the system in different basis. For example, if we project $|\psi \rangle$ onto position basis $\langle {\textbf {r}}|$, we would get $<{\textbf {r}}|\psi >\equiv \psi ({\textbf {r}})$, and if we project $|\psi \rangle$ onto momentum basis $\langle {\textbf {p}}|$, we would get $<{\textbf {p}}|\psi >\equiv \psi ({\textbf {p}})$, whereas $|\psi ({\textbf {r}})|^{2}$ is the probability of finding the system in the position ${\textbf {r}}$ and and $|\psi ({\textbf {p}})|^{2}$ is the probability of finding the system having the momentum ${\textbf {p}}$.

In Dirac notation, the scalar product of two state vectors ($\phi \$, $\psi \$) is denoted by a “bra-ket” $\langle \phi |\psi \rangle$. In coordinate representation the scalar product is given by:

$\langle \phi |\psi \rangle =\int \phi ^{*}(r,t)\psi (r,t)d^{3}r$

And so, the normalization condition may now be written:

$\langle \psi _{m}|\psi _{n}\rangle =\delta _{mn}$

Which additionally shows that any wavevector is determined to within a phase factor, $e^{i\gamma }$, where $\gamma$ is some real number.

The vectors in this space also obey some useful rules following from the fact that the Hilbert space is linear and complete:

$\langle \phi |c\psi \rangle =c\langle \phi |\psi \rangle$

$\langle c\phi |\psi \rangle =c^{*}\langle \phi |\psi \rangle$ where c is some c-number.

$\langle \phi |\psi _{1}+\psi _{2}\rangle =\langle \phi |\psi _{1}\rangle +\langle \phi |\psi _{2}\rangle$

In Dirac's notation, Schrödinger's equation is written as

$i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\mathcal {H}}|\psi (t)\rangle$

By projecting the equation in position space, we can obtain the previous form of Schrödinger's equation:

$i\hbar {\frac {\partial \psi ({\textbf {r}},t)}{\partial t}}=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\textbf {r}})\right]\psi ({\textbf {r}},t)$

On the other hand, we can also project the Schrödinger's equation in other space like momentum space and obtain:

$i\hbar {\frac {\partial \psi ({\textbf {p}},t)}{\partial t}}=\left[{\frac {{\textbf {p}}^{2}}{2m}}+V\left(i\hbar {\frac {\partial }{\partial {\textbf {p}}}}\right)\right]\psi ({\textbf {p}},t)$

where $\psi ({\textbf {p}},t)$ and $\psi ({\textbf {r}},t)$ are related through Fourier transform as described in the next section.

For time independent Hamiltonians, the Schrödinger's equation separates and we can seek the solution in the form of stationary states.

$|\psi _{n}(t)\rangle =e^{-iE_{n}t/\hbar }|\psi _{n}\rangle$.

The equation for stationary states in the Dirac notation is then

$E_{n}|\psi _{n}\rangle ={\mathcal {H}}|\psi _{n}\rangle .$

The eigenfunctions (now also referred to as eigenvectors) are replaced by eigenkets. Use of this notation makes solution of the Schrodinger equation much simpler for some problems, where the Hamiltonian can be re-written in the form of matrix operators having some algebra (defined set of operations on the basis vectors) over the Hilbert space of the eigenvectors of that Hamiltonian. (See the section on Operators.)

If we prepare an arbitrary state $|\phi \rangle$ at t=0, how does it evolve in time? $|\phi \rangle$ can be expressed as the linear superposition of the energy eignstates:

$|\phi \rangle =\sum _{n}c_{n}|\psi _{n}\rangle$

Then, we can get:

$|\phi (t)\rangle =e^{-i{\mathcal {H}}t/\hbar }|\phi \rangle =e^{-i{\mathcal {H}}t/\hbar }\sum _{n}c_{n}|\psi _{n}\rangle =\sum _{n}c_{n}e^{-iE_{n}t/\hbar }|\psi _{n}\rangle$.