How can I understand Schrodinger's wave equation
The Schrödinger equation
The Schrödinger equation - heuristic derivation
As already described, it is not possible to derive the Schrödinger equation mathematically.
Under one heuristic Derivation means an approach with which one can make the Schrödinger equation plausible for a special case. This is based on assumptions (which result from one or more theories), which ultimately prove to be correct in accordance with previous knowledge and experimental results.
We'll start with one one-dimensional wave function without external potential which reads analogously to the classical wave equation:
or written differently:
In this replace we: (Angular frequency) as (Wavenumber)
For the sake of simplicity, we set for the amplitude: . This results in:
As we have already shown (basics see above, Sections 3. / 4.), this wave function can alternatively be called complex exponential function write:
The sizes occurring in this function (Wavenumber) and (Angular frequency) should now with the help of the quantities pulse or. energy be expressed. To do this, we use the following relationships:
1) or. (DeBroglie)
With or. results for the pulse
and thus for the wave number
With and thus arises for the energy
and thus for the angular frequency
We now set the alternative expressions (1) and (2) into the wave function and get so
Deriving the wave function with respect to x
Now we derive this function (partially) from:
The factor on the right is equal to the wave function (so.).
In addition, the following applies: . This results in:
So with that
In words: The derivative of the wave function with respect to time multiplied by is equal to the product of momentum and the wave function.
Or in other words: applied to the wave function is equal to the wave function multiplied by the momentum.
That leads to
Replacement rule 1: or. ( = Momentum operator)
Deriving the wave function according to t
We now derive the wave function (partially) according to time from:
That leads to
Replacement rule 2: or. ( = Energy operator)
With the sizes energy and pulse on the right side it is about real sizesthat can be measured in an experiment.
On the left side there is one in each case Calculation rule (operator)how to and comes.
The physical quantities and the corresponding operators are assigned in quantum mechanics.
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