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Schrödinger equation

The Schrödinger equation, developed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. It is of central importance to the theory of quantum mechanics, playing a role analogous to Newton's second law in classical mechanics.

In quantum mechanics, the set of all possible states of a system is described by a complex Hilbert space, and any instantaneous state of a system is described by a unit vector in that space. This state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. 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.

Using Dirac's bra-ket notation, we denote that instantaneous state vector at time t by |ψ(t)⟩. The Schrödinger equation is:

$\mathbf{H} \left| \psi (t) \right\rangle = i \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle$

where i is the unit imaginary number, $\hbar$ is Planck's constant divided by 2π, and the Hamiltonian H is a Hermitian (self-adjoint) linear operator acting on the state space. The Hamiltonian describes the total energy of the system. As with the force occurring in Newton's second law, its exact form is not provided by the Schrödinger equation, and must be independently determined based on the physical properties of the quantum system.

For more information on the role of operators in quantum mechanics, see mathematical formulation of quantum mechanics.

For each Hamiltonian, there exists a set of quantum states, known as energy eigenstates, satisfying the eigenvalue equation

$\mathbf{H} |\psi(t)\rang = E |\psi(t)\rang$

Such a state possesses a definite total energy, whose value E is the eigenvalue of the state vector with the Hamiltonian. This eigenvalue equation is referred to as the time-independent Schrödinger equation. Hermitian 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.

On inserting the time-independent Schrödinger equation into the full Schrödinger equation,

$i \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle = E |\psi(t)\rang$

It is easy to solve this equation. We find that as time progresses, the state vectors of energy eigenstates change by only a complex phase:

$\left| \psi (t) \right\rangle = e^{-i E t / \hbar} |\psi(0)\rang$

Energy eigenstates are convenient to work with because their time-dependence is so simple; that is why the time-independent Schrödinger equation is so useful. We can always choose a set of instantaneous energy eigenstates whose state vectors {|n⟩} form a basis for the state space. Then any state vector |ψ(t)⟩ can be written as a linear superposition of energy eigenstates:

$|\psi(t)\rang = \sum_n c_n(t) |n\rang \quad,\quad \mathbf{H} |n\rang = E_n |n\rang \quad,\quad \sum_n |c_n(t)|^2 = 1$

(The last equation enforces the requirement that |ψ(t)⟩, like all state vectors, must be a unit vector.) Applying the Schrödinger equation to each side of the first equation, and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain

$i\hbar \frac{\partial c_n}{\partial t} = E_n c_n(t)$

Therefore, if we know the decomposition of |ψ(t)⟩ into the energy basis at time t = 0, its value at any subsequent time is given simply by

$|\psi(t)\rang = \sum_n e^{-iE_nt/\hbar} c_n(0) |n\rang$

The state space of many (but not all) quantum systems can be spanned with a position basis. In this situation, the Schrödinger equation may be conveniently reformulated as a partial differential equation for a wavefunction, a complex scalar field depending on position as well as time. This form of the Schrödinger equation is referred to as the Schrödinger wave equation.

Elements of the position basis are called position eigenstates. We will consider only a single-particle system, for which each position eigenstate may be denoted by |r⟩, where the label r is a real vector. This is to be interpreted as a state in which the particle is localized at position r. In this case, the state space is the space of all square-integrable complex functions.

The wavefunction

We define the wavefunction as the projection of the state vector |ψ(t)⟩ onto the position basis:

$\psi(\mathbf{r}, t) \equiv \left\langle \mathbf{r} | \psi(t) \right\rangle$

Since the position eigenstates form a basis for the state space, the integral over all projection operators is the identity operator:

$\int \left|\mathbf{r}\right\rangle \left\langle \mathbf{r} \right| d^3 \mathbf{r} = \mathbf{I}$

This statement is called the resolution of the identity[?]. With this, and the fact that kets have unit norm, we can show that

$\begin{matrix} 1 &=& \lang \psi(t) | \psi(t) \rang \\ &=& \lang \psi(t) | \; \left(\int \; |\mathbf{r}\rang \lang\mathbf{r}| \; d^3r \right) \; |\psi(t)\rang \\ &=& \int \; \lang\psi(t) |\mathbf{r}\rang \lang\mathbf{r}|\psi(t) \rang \; d^3 r \\ &=& \int \; \psi(\mathbf{r}, t)^* \; \psi(\mathbf{r}, t) \; d^3r \\ \end{matrix}$

where ψ(r, t)* denotes the complex conjugate of ψ(r, t). This important result tells us that the absolute square of the wavefunction, integrated over all space, must be equal to 1:

$\int \; |\psi(\mathbf{r}, t)|^2 \; d^3r = 1$

We can thus interpret the absolute square of the wavefunction as the probability density for the particle to be found at each point in space. In other words, |ψ(r, t)|² d³r is the probability, at time t, of finding the particle in the infinitesimal region of volume d³r surrounding the position r.

We have previously shown that energy eigenstates vary only by a complex phase as time progresses. Therefore, the absolute square of their wavefunctions do note change with time. Energy eigenstates thus correspond to static probability distributions.

Operators in the position basis

Operators A acting on the wavefunction are defined in the position basis by

$\mathbf{A} \psi(\mathbf{r}, t) \equiv \lang\mathbf{r}| \mathbf{A} | \psi(t) \rang$

The operators A on the two sides of the equation are different things: the one on the right acts on kets, whereas the one of the left acts on scalar fields. It is common to use the same symbols to denote operators acting on kets and their projections onto a basis. Usually, the kind of operator to which one is referring is apparent from the context, but this is a possible source of confusion.

The Schrödinger wave equation

Using the position-basis notation, the Schrödinger equation can be written in the position basis as:

$\mathbf{H} \psi(\mathbf{r},t) = i \hbar \frac{\partial}{\partial t} \psi(\mathbf{r},t)$

This form of the Schrödinger equation is the Schrödinger wave equation. It may appear that this is an ordinary differential equation, but in fact the Hamiltonian operator typically includes partial derivatives with respect to the position variable r. This usually leaves us with a difficult nonlinear partial differential equation to solve.

Often, the Hamiltonian can be expressed as the sum of two operators, one corresponding to kinetic energy and the other to potential energy. For a single particle of mass m with no electric charge and no spin, the kinetic energy term can be written as

$\frac{\left|\mathbf{p}\right|^2}{2m}$

where p is the momentum operator, defined by:

$\mathbf{p} \psi(\mathbf{r}, t) \equiv - i \hbar \nabla \psi(\mathbf{r}, t)$

The potential energy term can be expressed a real scalar function V = V(r), which describes the potential energy of the particle at position r. Putting these together, we obtain

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

where 2 is the Laplacian. This is a commonly encountered form of the Schrödinger wave equation, though not the most general one.

Analytical solutions of the time-independent Schrödinger equation can be obtained for a variety of relatively simple conditions. These solutions provide insight into the nature of quantum phenomena and sometimes provide a reasonable approximation of the behavior of more complex systems (e.g., in statistical mechanics, molecular vibrations are often approximated as harmonic oscillators). Several of the more common analytical solutions include:

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