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 Schrödinger equation is:
where:
Equation 28. Schrodinger equation
- is the reduced Planck constant
- is the wave function
- is the time
- is a linear operator called the Hamiltonian. It takes as input a function , 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 argument of could be anything, e.g.:Note however that there is always a single magical 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.
- 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. and for different particles. No matter how many particles there are, we have just a single , 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
The general Schrödinger 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.
- Non-relativistic quantum mechanics | 26, 5k, 66
- Quantum mechanics | 169, 20k, 270
- Particle physics | 137, 32k, 511
- Physics | 276, 47k, 835
- Natural science | 0, 64k, 2k
- Science | 0, 69k, 2k
- Ciro Santilli's Homepage | 262, 237k, 4k
- Conservation of the square amplitude in the Schrodinger equation | 66, 66, 1
- Derivation of the Klein-Gordon equation | 253
- Dirac equation | 329, 3k, 38
- Energy operator | 24, 52, 1
- History of quantum mechanics | 488, 488, 1
- Important partial differential equation | 43, 793, 24
- Klein-Gordon equation | 284, 537, 1
- Lagrangian mechanics | 907, 2k, 19
- Matrix mechanics | 225, 292, 2
- Non-relativistic quantum mechanics | 26, 5k, 66
- Particle physics | 137, 32k, 511
- Planck constant | 163, 195, 1
- Planck's law | 135
- Programmer's model of quantum computers
- Quantum electrodynamics | 682, 3k, 29
- Quantum mechanical re-interpretation of kinematic and mechanical relations by Heisenberg (1925) | 67
- Quantum mechanics | 169, 20k, 270
- Reduced Planck constant | 32
- Schrödinger equation | 555, 5k, 60
- Schrodinger equation for a one dimensional particle | 135
- Separation of variables | 82
- Spectral line | 165, 1k, 11
- Spin | 357, 994, 19
- Spontaneous emission | 91
- System of partial differential equations | 77
- Time-independent Schrodinger equation | 416
- Zeeman effect | 314