Stationary state

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A stationary state is a

Stationary states are quantum states that are solutions to the time-independent Schrödinger equation:

$${\displaystyle {\hat {H}}|\Psi \rangle =E_{\Psi }|\Psi \rangle ,}$$

This is an eigenvalue equation: ${\displaystyle {\hat {H}}}$ is a

${\displaystyle |\Psi \rangle }$

${\displaystyle {\hat {H}}}$

${\displaystyle E_{\Psi }}$

If a stationary state ${\displaystyle |\Psi \rangle }$ is plugged into the time-dependent Schrödinger equation, the result is^[2]

$${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\Psi \rangle =E_{\Psi }|\Psi \rangle .}$$

Assuming that ${\displaystyle {\hat {H}}}$ is time-independent (unchanging in time), this equation holds for any time t. Therefore, this is a differential equation describing how ${\displaystyle |\Psi \rangle }$ varies in time. Its solution is

$${\displaystyle |\Psi (t)\rangle =e^{-iE_{\Psi }t/\hbar }|\Psi (0)\rangle .}$$

Therefore, a stationary state is a standing wave that oscillates with an overall complex phase factor, and its oscillation angular frequency is equal to its energy divided by ${\displaystyle \hbar }$.

Stationary state properties

As shown above, a stationary state is not mathematically constant:

$${\displaystyle |\Psi (t)\rangle =e^{-iE_{\Psi }t/\hbar }|\Psi (0)\rangle .}$$

However, all observable properties of the state are in fact constant in time. For example, if ${\displaystyle |\Psi (t)\rangle }$ represents a simple one-dimensional single-particle wavefunction ${\displaystyle \Psi (x,t)}$, the probability that the particle is at location x is

$${\displaystyle |\Psi (x,t)|^{2}=\left|e^{-iE_{\Psi }t/\hbar }\Psi (x,0)\right|^{2}=\left|e^{-iE_{\Psi }t/\hbar }\right|^{2}\left|\Psi (x,0)\right|^{2}=\left|\Psi (x,0)\right|^{2},}$$

The Heisenberg picture is an alternative mathematical formulation of quantum mechanics where stationary states are truly mathematically constant in time.

As mentioned above, these equations assume that the Hamiltonian is time-independent. This means simply that stationary states are only stationary when the rest of the system is fixed and stationary as well. For example, a 1s electron in a hydrogen atom is in a stationary state, but if the hydrogen atom reacts with another atom, then the electron will of course be disturbed.

Spontaneous decay

Spontaneous decay complicates the question of stationary states. For example, according to simple (

An orbital is a stationary state (or approximation thereof) of a one-electron atom or molecule; more specifically, an atomic orbital for an electron in an atom, or a molecular orbital for an electron in a molecule.^[4]

For a molecule that contains only a single electron (e.g. atomic

In chemistry, calculation of molecular orbitals typically also assume the Born–Oppenheimer approximation.

See also

• Transition of state
• Quantum number
• vacuum state
• Virtual particle
• Steady state

References