Materials Science and Engineering/List of Topics/Quantum Mechanics/Schrodinger Equation

In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to Newton's second law in classical mechanics for macroscopic particles. Microscopic particles include elementary particles, such as electrons, as well as systems of particles, such as atomic nuclei. Macroscopic particles vary in mass from cells to the galactic superclusters.

Historical background and development

Schrödinger's equation follows very naturally from earlier developments:

In 1905, by considering the photoelectric effect, Albert Einstein had published his

${\displaystyle E=hf\;}$

formula for the relation between the energy E and frequency f of the quanta of radiation (photons), where h is Planck's constant.

In 1924 Louis de Broglie presented his de Broglie hypothesis which states that all particles (not just photons) have an associated wavefunction ${\displaystyle \Psi \;}$ with properties:

${\displaystyle p=h/\lambda \;}$, where ${\displaystyle \lambda \,}$ is the wavelength of the wave and p the momentum of the particle.

De Broglie showed that this was consistent with Einstein's formula and special relativity so that

${\displaystyle E=hf\;}$

still holds, but now this is hypothesized to hold for all particles, not just photons anymore.

Expressed in terms of angular frequency ${\displaystyle \omega =2\pi f\;}$ and wavenumber ${\displaystyle k=2\pi /\lambda \;}$, with ${\displaystyle \hbar =h/2\pi \;}$ we get:

${\displaystyle E=\hbar \omega }$

and

${\displaystyle \mathbf {p} =\hbar \mathbf {k} \;}$

where we have expressed p and k as vectors.

Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor:

${\displaystyle \psi \approx e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}}$

and to realize that since

${\displaystyle {\frac {\partial }{\partial t}}\psi =-i\omega \psi }$

then

${\displaystyle E\psi =\hbar \omega \psi =i\hbar {\frac {\partial }{\partial t}}\psi }$

and similarly since:

${\displaystyle {\frac {\partial }{\partial x}}\psi =ik_{x}\psi }$

then

${\displaystyle p_{x}\psi =\hbar k_{x}\psi =-i\hbar {\frac {\partial }{\partial x}}\psi }$

and hence:

${\displaystyle p_{x}^{2}\psi =-\hbar ^{2}{\frac {\partial ^{2}}{\partial x^{2}}}\psi }$

so that, again for a plane wave, he got:

${\displaystyle p^{2}\psi =(p_{x}^{2}+p_{y}^{2}+p_{z}^{2})\psi =-\hbar ^{2}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)\psi =-\hbar ^{2}\nabla ^{2}\psi }$

And by inserting these expressions into the Newtonian formula for a particle with total energy E, mass m, moving in a potential V:

${\displaystyle E={\frac {p^{2}}{2m}}+V}$ (simply the sum of the kinetic energy and potential energy; the plane wave model assumed V = 0)

he got his famed equation for a single particle in the 3-dimensional case in the presence of a potential:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi }$

Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, ${\displaystyle \psi \;}$, moving in a potential well, V, created by the positively charged proton. This computation tallied with experiment, the Bohr model and also the results of Werner Heisenberg's matrix mechanics - but without having to introduce Heisenberg's concept of non-commuting observables. Schrödinger published his wave equation and the spectral analysis of hydrogen in a series of four papers in 1926.

The Schrödinger equation defines the behaviour of ${\displaystyle \psi \;}$, but does not interpret what ${\displaystyle \psi \;}$ is. Schrödinger tried unsuccessfully to interpret it as a charge density. In 1926 Max Born, just a few days after Schrödinger's fourth and final paper was published, successfully interpreted ${\displaystyle \psi \;}$ as a probability amplitude, although Schrödinger was never reconciled to this statistical or probabilistic approach.

Mathematical formulation

In the mathematical formulation of quantum mechanics, a physical system is associated with a complex Hilbert space such that each instantaneous state of the system is described by a ray in that space. The nonzero elements of a Hilbert space are by definition normalizable and it is convenient, although not necessary, to represent a state by an element of the ray which is normalized to unity. This vector is often somewhat loosely referred to as wave function, although in a more rigorous formulation of quantum mechanics a wave function is a special case of a state vector. (In fact, a wave function is a state in the position representation, see below). A state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. It contains all information of the system that is knowable in a quantum mechanical sense. As the state of a system generally changes over time, the state vector is a function of time. The Schrödinger equation provides a quantitative description of the rate of change of the state vector.

In Dirac's bra-ket notation at time ${\displaystyle t}$ the state is given by the ket ${\displaystyle |\psi (t)\rangle }$. The time-dependent Schrödinger equation, giving the time evolution of the ket, is:

${\displaystyle H(t)\left|\psi \left(t\right)\right\rangle =\mathrm {i} \hbar {\frac {d}{dt}}\left|\psi \left(t\right)\right\rangle }$

where ${\displaystyle \mathrm {i} }$ is the imaginary unit, ${\displaystyle t}$ is time, ${\displaystyle d/dt}$ is the derivative with respect to ${\displaystyle t}$, ${\displaystyle \hbar }$ is the reduced Planck's constant (Planck's constant divided by ${\displaystyle 2\pi \,}$), ${\displaystyle \psi (t)\,}$ is the time dependent state vector, and ${\displaystyle H(t)}$ is the Hamiltonian (a self-adjoint operator acting on the state space). If one assumes a certain representation for ${\displaystyle \psi \,}$, for instance position or momentum representation, the state vector is assumed to depend on more variables than time alone, and the time derivative must be replaced by the partial derivative ${\displaystyle \partial /\partial t.}$

The Hamiltonian describes the total energy of the system. As with the force occurring in Newton's second law, its form is not provided by the Schrödinger equation, but must be independently determined from the physical properties of the system.

Time-independent Schrödinger equation

For many real-world problems the energy operator (${\displaystyle H}$), does not depend on time. In such problems, there may exist a static, or time independent solution. For such a solution the time-dependent Schrödinger equation simplifies[1] to the time-independent Schrödinger equation, which has the well-known appearance ${\displaystyle H\Psi =E\Psi \,}$.

An example of a simple one-dimensional time-independent Schrödinger equation for a particle of mass m, moving in a potential U(x) is: [1]

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+U(x)\psi (x)=E\psi (x).}$

The analogous 3-dimensional time-independent equation is, [2]:

${\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+U(\mathbf {r} )\right]\psi (\mathbf {r} )=E\psi (\mathbf {r} ),}$

or

${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +(U-E)\psi =0,}$

where ${\displaystyle \nabla }$ is the del operator.

For every time-independent Hamiltonian, ${\displaystyle H}$, there exists a set of quantum states, ${\displaystyle \left|\psi _{n}\right\rangle }$, known as energy eigenstates, and corresponding real numbers ${\displaystyle E_{n}}$ satisfying the eigenvalue equation,

${\displaystyle H\left|\psi _{n}\right\rangle =E_{n}\left|\psi _{n}\right\rangle .}$

Such a state possesses a definite total energy, whose value ${\displaystyle E_{n}}$ is the eigenvalue of the Hamiltonian. The corresponding eigenvector ${\displaystyle \psi _{n}\,}$ is normalizable to unity. This eigenvalue equation is referred to as the time-independent Schrödinger equation. We purposely left out the variable(s) on which the wavefunction ${\displaystyle \psi _{n}\,}$ depends. In the first example above it depends on the single variable x and in the second on x, y, and z—the components of the vector r. In both cases the Schrödinger equation has the same appearance, but its Hamilton operator is defined on different function (state, Hilbert) spaces. In the first example the function space consists of functions of one variable and in the second example the function space consists of functions of three variables.

Self-adjoint operators, such as the Hamiltonian, have the property that their eigenvalues are always real numbers, as we would expect, since the energy is a physically observable quantity. Sometimes more than one linearly independent state vector correspond to the same energy ${\displaystyle E_{n}}$. If the maximum number of linearly independent eigenvectors corresponding to ${\displaystyle E_{n}}$ equals k, we say that the energy level ${\displaystyle E_{n}}$ is k-fold degenerate. When k=1 the energy level is called non-degenerate.

On inserting a solution of the time-independent Schrödinger equation into the full Schrödinger equation, we get

${\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}\left|\psi _{n}\left(t\right)\right\rangle =E_{n}\left|\psi _{n}\left(t\right)\right\rangle .}$

It is relatively easy to solve this equation. One finds that the energy eigenstates (i.e., solutions of the time-independent Schrödinger equation) change as a function of time only trivially, namely, only by a complex phase:

${\displaystyle \left|\psi \left(t\right)\right\rangle =\mathrm {e} ^{-\mathrm {i} Et/\hbar }\left|\psi \left(0\right)\right\rangle .}$

It immediately follows that the probability amplitude,

${\displaystyle \psi (t)^{*}\psi (t)=\mathrm {e} ^{\mathrm {i} Et/\hbar }\mathrm {e} ^{-\mathrm {i} Et/\hbar }\psi (0)^{*}\psi (0)=|\psi (0)|^{2},}$

is time-independent. Because of a similar cancellation of phase factors in bra and ket, all average (expectation) values of time-independent observables (physical quantities) computed from ${\displaystyle \psi (t)\,}$ are time-independent.

Energy eigenstates are convenient to work with because they form a complete set of states. That is, the eigenvectors ${\displaystyle \left\{\left|n\right\rangle \right\}}$ form a basis for the state space. We introduced here the short-hand notation ${\displaystyle |\,n\,\rangle =\psi _{n}}$. Then any state vector that is a solution of the time-dependent Schrödinger equation (with a time-independent ${\displaystyle H}$) ${\displaystyle \left|\psi \left(t\right)\right\rangle }$ can be written as a linear superposition of energy eigenstates:

${\displaystyle \left|\psi \left(t\right)\right\rangle =\sum _{n}c_{n}(t)\left|n\right\rangle \quad ,\quad H\left|n\right\rangle =E_{n}\left|n\right\rangle \quad ,\quad \sum _{n}\left|c_{n}\left(t\right)\right|^{2}=1.}$

(The last equation enforces the requirement that ${\displaystyle \left|\psi \left(t\right)\right\rangle }$, like all state vectors, may be normalized to a unit vector.) Applying the Hamiltonian operator to each side of the first equation, the time-dependent Schrödinger equation in the left-hand side and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain

${\displaystyle \mathrm {i} \hbar {\frac {\partial c_{n}}{\partial t}}=E_{n}c_{n}\left(t\right).}$

Therefore, if we know the decomposition of ${\displaystyle \left|\psi \left(t\right)\right\rangle }$ into the energy basis at time ${\displaystyle t=0}$, its value at any subsequent time is given simply by

${\displaystyle \left|\psi \left(t\right)\right\rangle =\sum _{n}\mathrm {e} ^{-\mathrm {i} E_{n}t/\hbar }c_{n}\left(0\right)\left|n\right\rangle .}$

Note that when some values ${\displaystyle c_{n}(0)\,}$ are not equal to zero for differing energy values ${\displaystyle E_{n}\,}$, the left-hand side is not an eigenvector of the energy operator ${\displaystyle H}$. The left-hand is an eigenvector when the only ${\displaystyle c_{n}(0)\,}$-values not equal to zero belong the same energy, so that ${\displaystyle \mathrm {e} ^{-\mathrm {i} E_{n}t/\hbar }}$ can be factored out. In many real-world application this is the case and the state vector ${\displaystyle \psi (t)\,}$ (containing time only in its phase factor) is then a solution of the time-independent Schrödinger equation.

Example

Let ${\displaystyle |\,1\,\rangle }$ and ${\displaystyle |\,2\,\rangle }$ be degenerate eigenstates of the time-independent Hamiltonian ${\displaystyle H\,}$:

${\displaystyle H\,|\,1\,\rangle =E|\,1\,\rangle \quad {\hbox{and}}\quad H\,|\,2\,\rangle =E|\,2\,\rangle .}$

Suppose a solution ${\displaystyle \psi (t)\,}$ of the full (time-dependent) Schrödinger equation of ${\displaystyle H\,}$ has the form at t = 0:

${\displaystyle |\,\psi (0)\,\rangle =c_{1}|\,1\,\rangle +c_{2}|\,2\,\rangle .}$

Hence, because of the discussion above, at t > 0 :

${\displaystyle |\,\psi (t)\,\rangle =\mathrm {e} ^{-\mathrm {i} Et/\hbar }c_{1}|\,1\,\rangle +\mathrm {e} ^{-\mathrm {i} Et/\hbar }c_{2}|\,2\,\rangle =\mathrm {e} ^{-\mathrm {i} Et/\hbar }\left(c_{1}|\,1\,\rangle +c_{2}|\,2\,\rangle \right)=\mathrm {e} ^{-\mathrm {i} Et/\hbar }|\,\psi (0)\,\rangle ,}$

which shows that ${\displaystyle \psi (t)\,}$ only depends on time in a trivial way (through its phase), also in the case of degeneracy.

Apply now ${\displaystyle H\,}$:

${\displaystyle H\,|\,\psi (t)\,\rangle =\mathrm {e} ^{-\mathrm {i} Et/\hbar }c_{1}E\,|\,1\,\rangle +\mathrm {e} ^{-\mathrm {i} Et/\hbar }c_{2}E\,|\,2\,\rangle =E\mathrm {e} ^{-\mathrm {i} Et/\hbar }\left(c_{1}|\,1\,\rangle +c_{2}|\,2\,\rangle \right)}$
${\displaystyle =E\mathrm {e} ^{-\mathrm {i} Et/\hbar }|\,\psi (0)\,\rangle =E\,|\,\psi (t)\,\rangle .}$

Conclusion: The wavefunction ${\displaystyle \psi (t)\,}$ with the given initial condition (its form at t = 0), remains a solution of the time-independent Schrödinger equation ${\displaystyle H\psi (t)=E\psi (t)}$ for all times t > 0.

References

1. In fact also an initial condition must be used here. At time zero the wavefunction must be an eigenstate of ${\displaystyle H.}$