Matching a wavefunction and its derivative across finite potential discontinuities, together with decay or wall conditions, permits only discrete bound-state energies.

Time-dependent Schrodinger equation ( $i ℏ \partial_{t} ψ = H ψ$ )

 0  0

The time-dependent Schrodinger equation evolves a quantum state according to

i ℏ \frac{\partial ψ}{\partial t} = H ψ .

(1)

Madelung equations ( $ψ = R e^{i S}$ )

 1  0

Writing the wavefunction in modulus and dimensionless quantum phase gives a probability continuity equation and a Hamilton-Jacobi equation with a quantum potential:

\partial_{t} (R^{2}) + \nabla \cdot (R^{2} \frac{ℏ}{m} \nabla S) = 0, ℏ S_{t} + \frac{ℏ ^{2}}{2 m} ∣\nabla S ∣^{2} + V - \frac{ℏ ^{2}}{2 m} \frac{\nabla ^{2} R}{R} = 0.

(1)

The phase equations hold locally away from wavefunction nodes. They follow by equating real and imaginary parts of the Time-dependent Schrodinger equation.

Nondegeneracy of one-dimensional bound states

 0  0

Two bound-state solutions of the same one-dimensional time-independent Schrodinger equation have constant Wronskian. Their decay at infinity makes this constant zero, so the two solutions are linearly dependent. Every bound-state energy is therefore nondegenerate.

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This is a section about Schrödinger equation!

Schrödinger equation is a very important subject about which there is a lot to say.

For example, this sentence. And then another one.

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Schrödinger equation by

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The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is a key principle in understanding wave functions and the behavior of particles at the quantum level. There are two forms of the Schrödinger equation: 1. **Time-dependent Schrödinger equation**: This form is used to describe how the quantum state evolves over time.

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Schrödinger equation by

Ciro Santilli  40 Updated 2025-07-16

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The partial differential equation of non-relativistic quantum mechanics.

Experiments explained:

via the Schrödinger equation solution for the hydrogen atom it predicts:
- spectral line basic lines, plus Zeeman effect
Schrödinger equation solution for the helium atom: perturbative solutions give good approximations to the energy levels
double-slit experiment: I think we have a closed solution for the max and min probabilities on the measurement wall, and they match experiments

Experiments not explained: those that the Dirac equation explains like:

To get some intuition on the equation on the consequences of the equation, have a look at:

The easiest to understand case of the equation which you must have in mind initially that of the Schrödinger equation for a free one dimensional particle.

Then, with that in mind, the general form of the Schrödinger equation is:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = \hat{H} [ψ (x, t)]

(1)

Equation 1.

Schrodinger equation

where:

$ℏ$ is the reduced Planck constant
$ψ$ is the wave function
$t$ is the time
$\hat{H}$ is a linear operator called the Hamiltonian. It takes as input a function $ψ$ , and returns another function. This plays a role analogous to the Hamiltonian in classical mechanics: determining it determines what the physical system looks like, and how the system evolves in time, because we can just plug it into the equation and solve it. It basically encodes the total energy and forces of the system.

The

x

argument of

ψ

could be anything, e.g.:

we could have preferred polar coordinates instead of linear ones if the potential were symmetric around a point
we could have more than one particle, e.g. solutions of the Schrodinger equation for two electrons, which would have e.g. $x_{1}$ and $x_{2}$ for different particles. No matter how many particles there are, we have just a single $ψ$ , we just add more arguments to it.
we could have even more generalized coordinates. This is much in the spirit of Hamiltonian mechanics or generalized coordinates

Note however that there is always a single magical

t

time variable. This is needed in particular because there is a time partial derivative in the equation, so there must be a corresponding time variable in the function. This makes the equation explicitly non-relativistic.

The general Schrödinger equation can be broken up into a trivial time-dependent and a time-independent Schrödinger equation by separation of variables. So in practice, all we need to solve is the slightly simpler time-independent Schrödinger equation, and the full equation comes out as a result.

Existence and uniqueness: mathoverflow.net/questions/212913/existence-and-uniqueness-for-two-dimensional-time-dependent-schr%C3%B6dinger-equation

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