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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( |\psi_0\rangle, |\psi_1\rangle, |\psi_2\rangle, ... , |\psi_j\rangle, ... \right) } , where each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_i\rangle } is a complex vector function.
We denote a state vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\ } in Hilbert space with Dirac notation as a “ket” Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \psi \rangle} , and its complex conjugate (or dual vector) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\ } * is denoted by a “bra” Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \phi \rangle = \sum_n c_n|\psi_n\rangle } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_n } denotes a complex number.
By projecting the state vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle}
onto different basis, we can obtain the wavefunctions of the system in different basis. For example, if we project Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle}
onto position basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \textbf{r}|}
, we would get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\textbf{r}|\psi> \equiv \psi(\textbf{r})}
, and if we project Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle}
onto momentum basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \textbf{p}|}
, we would get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\textbf{p}|\psi> \equiv \psi(\textbf{p})}
, whereas Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi( \textbf{r} )|^{2}}
is the probability of finding the system in the position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{r}}
and and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi( \textbf{p} )|^{2}}
is the probability of finding the system having the momentum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{p}}
.
In Dirac notation, the scalar product of two state vectors (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi\ }
, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\ }
) is denoted by a “bra-ket” Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\phi|\psi\rangle }
. In coordinate representation the scalar product is given by:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\phi|\psi\rangle = \int \phi^*(r,t)\psi(r,t)d^3r }
And so, the normalization condition may now be written:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi_m|\psi_n\rangle = \delta_{mn} }
Which additionally shows that any wavevector is determined to within a phase factor, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\gamma}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\phi|c\psi\rangle = c\langle\phi|\psi\rangle }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle c\phi|\psi\rangle = c^*\langle\phi|\psi\rangle } where c is some c-number.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(\textbf{p},t)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_n(t)\rangle=e^{-iE_n t/\hbar}|\psi_n\rangle} .
The equation for stationary states in the Dirac notation is then
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Schrödinger 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle } at t=0, how does it evolve in time? Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle } can be expressed as the linear superposition of the energy eignstates:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \phi \rangle=\sum_{n}c_n| \psi_n \rangle }
Then, we can get:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \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_ne^{-iE_n t/\hbar}|\psi_n\rangle } .