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. More specifically, it is an equation describing the components of the state vector in position space. We will now introduce this more general equation, written in terms of the state vector itself, 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}+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.
With the aid of the identity,
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 \frac{\partial}{\partial x}\delta(x'-x)=\frac{\delta(x'-x)}{x'-x},}
one may verify that
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'|\hat{p}|x\rangle=i\hbar\frac{\delta(x'-x)}{x'-x}=i\hbar\frac{\partial}{\partial x}\delta(x'-x)}
and that
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{p}|\Psi(t)\rangle=-i\hbar\int dx\,\frac{\partial\Psi}{\partial x}|x\rangle.}
If we now substitute the above form of the Hamiltonian into the Schrödinger equation and project the resulting equation into position space, we will arrive at the wave equation stated in the previous chapter,
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(x,t)}{\partial t}=\left [-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x,t)\right ]\Psi(x,t).}
The above procedure can be generalized to multiple dimensions, again recovering the multi-dimensional wave equation given in the previous chapter:
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)}
We could also have chosen to work in momentum space; a similar procedure yields
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).}
Here, 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 a Fourier transform as described in a previous section.
Let us now consider a time-independent Hamiltonian. As described previously, we may solve the Schrödinger equation in this case by first assuming that the state vector has the form,
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,}
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_n\rangle} is independent of time. Substituting this form into the Schrödinger equation yields 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=\hat{H}|\psi_n\rangle.}
The eigenfunctions are replaced with eigenvectors. Use of this notation makes solution of the Schrödinger equation much simpler for some problems; if we write the eigenvectors in a convenient basis, we may project the above eigenvalue equation onto all states in the basis, thus reducing the problem to diagonalizing a matrix.
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 |\Psi(t)\rangle} evolves in time for a time-independent Hamiltonian. Let us expand this state in terms of an orthonormal 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 |1\rangle,|2\rangle,\ldots,} obtaining
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=\sum_{n}c_{n}(t)|n\rangle.}
If we now substitute this into the Schrödinger equation and project the result onto each of the basis states, we 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{dc_{m}(t)}{dt}=\sum_{mn}\langle m|H|n\rangle c_{n}(t).}
If, in particular, we work in the 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 \left\{|\psi_n\rangle\right\}} of eigenstates of the Hamiltonian, the above equation reduces to a set of decoupled equations for the coefficient of each eigenstate,
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{dc_{m}(t)}{dt}=E_{m}c_{m}(t),}
whose solutions are 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_{m}(t)=c_{m}(0)e^{-iE_{m}t/\hbar}.} Therefore, the time evolution of a general state 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 |\Psi(t)\rangle=\sum_{n}c_{n}(0)e^{-iE_{n}t/\hbar}|\psi_n\rangle,}
which is just a linear superposition of the time-dependent eigenvectors obtained previously. One could also have obtained this from the fact that any linear superposition of solutions of the Schrödinger equation is itself a solution, and thus any state may be written in the above form.
We will now show that the Schrödinger equation in this form preserves the normalization of the state vector; i.e., if the vector is normalized initially, then it will remain normalized at all times. We start by writing the dual 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{d}{dt}\langle\Psi(t)|=\langle\Psi(t)|\hat{H}(t).}
We now act on the Schrödinger equation the left with 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(t)|} and on its dual from the right with 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} and subtract the two results, obtaining
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(t)|\frac{d}{dt}|\Psi(t)\rangle+\frac{d}{dt}\left [\langle\Psi(t)|\right ]|\Psi(t)\rangle=0,}
or
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 \frac{d}{dt}\langle\Psi(t)|\Psi(t)\rangle=0.}
As asserted, 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(t)|\Psi(t)\rangle=\text{const.},} so that we only need to normalize the state vector at 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 t=0.\!}