Dynamics related to single particle in a potential / other spatial properties

In this situation, the SE is given by the form

$${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi _{\alpha }(\mathbf {r} ,\,t)={\hat {H}}\Psi _{\alpha }(\mathbf {r} ,\,t)=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\right)\Psi _{\alpha }(\mathbf {r} ,\,t)=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi _{\alpha }(\mathbf {r} ,\,t)+V(\mathbf {r} )\Psi _{\alpha }(\mathbf {r} ,\,t)}$$

It can be derived from (1) by considering ${\displaystyle \Psi _{\alpha }(x,t):=\langle x|\alpha \rangle }$ and ${\displaystyle {\hat {H}}:=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\hat {V}}}$

Bound state

A state is called bound state if its position probability density at infinite tends to zero for all the time. Roughly speaking, we can expect to find the particle(s) in a finite size region with certain probability. More precisely, ${\displaystyle |\psi (\mathbf {r} ,t)|^{2}\to 0}$ when ${\displaystyle |\mathbf {r} |\to +\infty }$, for all ${\displaystyle t>0}$.

There is a criterion in terms of energy:

Let ${\displaystyle E}$ be the expectation energy of the state. It is a bound state if and only if ${\displaystyle E<\operatorname {min} \{V(r\to -\infty ),V(r\to +\infty )\}}$.

Position representation and momentum representation

Position representation of a wave function

${\displaystyle \Psi _{\alpha }(x,t):=\langle x|\alpha \rangle }$,

momentum representation of a wave function

${\displaystyle {\tilde {\Psi }}_{\alpha }(p,t):=\langle p|\alpha \rangle }$ ;

where ${\displaystyle |x\rangle }$ is the position eigenstate and ${\displaystyle |p\rangle }$ the momentum eigenstate respectively.

The two representations are linked by Fourier transform.

Probability amplitude

A probability amplitude is of the form ${\displaystyle \langle \alpha |\psi \rangle }$.

Probability current

Having the metaphor of probability density as mass density, then probability current ${\displaystyle J}$ is the current: ${\displaystyle J(x,t)={\frac {i\hbar }{2m}}\left(\psi {\frac {\partial \psi ^{*}}{\partial x}}-{\frac {\partial \psi }{\partial x}}\psi \right)}$ The probability current and probability density together satisfy the continuity equation:

$${\displaystyle {\frac {\partial }{\partial t}}|\psi (x,t)|^{2}+\nabla \cdot \mathbf {J} (x,t)=0}$$

Probability density

Given the wave function of a particle, ${\displaystyle |\psi (x,t)|^{2}}$ is the probability density at position ${\displaystyle x}$ and time ${\displaystyle t}$. ${\displaystyle |\psi (x_{0},t)|^{2}\,dx}$ means the probability of finding the particle near ${\displaystyle x_{0}}$.

Scattering state

The wave function of scattering state can be understood as a propagating wave. See also "bound state".

There is a criterion in terms of energy:

Let ${\displaystyle E}$ be the expectation energy of the state. It is a scattering state if and only if ${\displaystyle E>\operatorname {min} \{V(r\to -\infty ),V(r\to +\infty )\}}$.

Square-integrable

Square-integrable is a necessary condition for a function being the position/momentum representation of a wave function of a bound state of the system.

Given the position representation ${\displaystyle \Psi (x,t)}$ of a state vector of a wave function, square-integrable means:

• 1D case: ${\displaystyle \int _{-\infty }^{+\infty }|\Psi (x,t)|^{2}\,dx<+\infty }$.
• 3D case: ${\displaystyle \int _{V}|\Psi (\mathbf {r} ,t)|^{2}\,dV<+\infty }$.

Stationary state

A stationary state of a bound system is an eigenstate of Hamiltonian operator. Classically, it corresponds to standing wave. It is equivalent to the following things:^{[nb 2]}

• an eigenstate of the Hamiltonian operator
• an eigenfunction of Time-Independent Schrödinger Equation
• a state of definite energy
• a state which "every expectation value is constant in time"
• a state whose probability density (${\displaystyle |\psi (x,t)|^{2}}$) does not change with respect to time, i.e. ${\displaystyle {\frac {d}{dt}}|\Psi (x,t)|^{2}=0}$