The Schrödinger Equation in Dirac Notation
The Schrödinger equation, as introduced in the previous chapter, is a special case of a more general equation that is satisfied by the abstract 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(t)\rangle} describing the system. We will now introduce this more general equation, and show how one can recover the wave equation from the previous chapter.
In Dirac notation, the Schrödinger 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{d}{dt}|\Psi(t)\rangle=\hat{H}(t)|\Psi(t)\rangle.}
We see that the Hamiltonian of the system determines how a given initial state will evolve in time.
To show how to recover the equation for the wave function, let us consider the Hamiltonian for a particle moving in one dimension,
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 \hat{H}=\frac{\hat{p}^2}{2m}+\hat{V}(\hat{x},t).}
We now write our state vector in position space. Since the position space is continuous, rather than discrete, the state vector as a linear superposition of position eigenstates must now be written as an integral:
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(t)\rangle=\int dx\,\Psi(x,t)|x\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 \langle x|x'\rangle=\delta(x-x')} 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 \delta(x)\!} is the Dirac delta function.
By projecting the equation in position space, we can obtain the previous form of the Schrödinger 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 it into 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 \phi(\textbf{p},t)}{\partial t} = \left[ \frac{\textbf {p}^{2}}{2m} + V\left ( i\hbar \frac{\partial}{\partial \textbf{p}}\right)\right]\phi(\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 \phi(\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 wave function may be separated into a position-dependent part and a time-dependent part,
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} .
as described previously, thus yielding the equation for stationary states in Dirac notation:
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.)
We now ask how 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 } evolves in time? The initial 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 } 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 }
We can then solve the time-dependent Schrödinger equation, we obtain, for a time-independent Hamiltonian,
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.}