Schrodinger equation

Tags: Important partial differential equation

🔗

The partial differential equation of non-relativistic quantum mechanics.

🔗

Experiments explained:

via the Schrodinger equation solution for the hydrogen atom it predicts:
- spectral line basic lines, plus Zeeman effect
Schrodinger 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:

fine structure
spontaneous emission coefficients

🔗

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 Schrodinger equation for a free one dimensional particle.

🔗

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

Equation 26. Schrodinger equation

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

(26)

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

🔗

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

🔗

Table of contents | 555, 4k, 55
- 1. Time-independent Schrodinger equation | 381 | 🔗 link | 🗖 nosplit | ↑ parent "Schrodinger equation"
- 2. Derivation of the Schrodinger equation | 99, 152, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Schrodinger equation"
  - 2.1. Why are complex numbers used in the Schrodinger equation? | 53 | 🔗 link | 🗖 nosplit | ↑ parent "Derivation of the Schrodinger equation"
- 3. Solutions of the Schrodinger equation | 34, 2k, 34 | 🔗 link | 🗖 nosplit | ↑ parent "Schrodinger equation"
- 4. Born-Oppenheimer approximation | 🔗 link | 🗖 nosplit | ↑ parent "Schrodinger equation"
- 5. Uncertainty principle | 292, 1k, 10 | 🔗 link | 🗖 nosplit | ↑ parent "Schrodinger equation"
  - 5.1. Position and momentum space | 187, 435, 4 | 🔗 link | 🗖 nosplit | ↑ parent "Uncertainty principle"
  - 5.2. Energy operator | 24, 52, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Uncertainty principle"
    - 5.2.1. Time-energy uncertainty principle | 28 | 🔗 link | 🗖 nosplit | ↑ parent "Energy operator"
  - 5.3. Angular momentum operator | 217, 226, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Uncertainty principle"
    - 5.3.1. Total angular momentum operator | 9 | 🔗 link | 🗖 nosplit | ↑ parent "Angular momentum operator"
  - 5.4. Complementarity (physics) | 🔗 link | 🗖 nosplit | ↑ parent "Uncertainty principle"
- 6. Conservation laws in Schrodinger equations | 20, 86, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Schrodinger equation"
  - 6.1. Conservation of the square amplitude in the Schrodinger equation | 66, 66, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Conservation laws in Schrodinger equations"
    - 6.1.1. Probability current | 🔗 link | 🗖 nosplit | ↑ parent "Conservation of the square amplitude in the Schrodinger equation"
- 7. Wave function ( $ψ$ ) | 107, 107, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Schrodinger equation"
  - 7.1. Matter wave | 🔗 link | 🗖 nosplit | ↑ parent "Wave function"

🔗

Ancestors

Non-relativistic quantum mechanics | 26, 4k, 59
Quantum mechanics | 170, 18k, 243
Particle physics | 137, 28k, 458
Physics | 276, 40k, 705
Natural science | 0, 49k, 1k
Science | 0, 53k, 1k
Ciro Santilli's Homepage | 262, 182k, 3k

🔗

Incoming links

Conservation of the square amplitude in the Schrodinger equation | 66, 66, 1
Derivation of the Klein-Gordon | 253
Dirac equation | 329, 2k, 29
Energy operator | 24, 52, 1
Important partial differential equation | 43, 793, 24
Klein-Gordon equation | 243, 496, 1
Lagrangian mechanics | 907, 2k, 19
Matrix mechanics | 186
Non-relativistic quantum mechanics | 26, 4k, 59
Particle physics | 137, 28k, 458
Planck constant | 163, 258, 2
Planck's law | 118, 452, 4
Quantum electrodynamics | 678, 3k, 28
Quantum mechanics | 170, 18k, 243
Reduced Planck constant | 11
Schrodinger equation | 555, 4k, 55
Schrodinger equation for a one dimensional particle | 135
Separation of variables | 82
Spectral line | 165, 1k, 10
Spin | 229, 596, 13
Spontaneous emission | 91
System of partial differential equations | 77
Time-independent Schrodinger equation | 381
Zeeman effect | 314