diff --git a/Wavefunction.txt b/Wavefunction.txt new file mode 100644 index 0000000..093837f --- /dev/null +++ b/Wavefunction.txt @@ -0,0 +1,12 @@ +[t] As an introduction to the wavefunction we may have a quick reconciliation on the wave-model. So the first part that was introduced was the wavelength, it basically meant, that there was an uncertainty of the position of the quantum object, the physical meaning of this concept is that physicists of that time thought that the quantum object was in a sphere defined by the wavelength. However now things have changed, into both directions. \ +On one side we now can give clearer information about the position of the quantum object in the same sphere. On the other side we have clarified that there is a small but existing probability of the quantum object being outside of that sphere. \ +The wavefunction is usually written as Ψ(x, y, z, t) or just as a function of a selected amount of those variables. It is the core of quantum theory and it is the function wich is the solution to the Schroedinger-equation, which is of course a differential equation so it has a function as a solution. One can express it's meaning very simply by stating, that: [\] + +[f] | Ψ(x, y, z, t)|^2 = p(x, y, z, t) [\] + +[t] In this context p is the probability of finding a particle at a certain spot in the space-time-continuum. As you might have noticed by now (while reading articles of this app) quantum physics is non-deterministic. So there is no way for you, or anybody to calculate where exactly a particle (or other quantum object) is going to be. And that has nothing to do with a lack of information. \ +As far as I am concerned there is just no way to know this. The problem is the only thing the equations of quantum physics (mainly the Schroedinger-equation) do not tell us exactly how the universe is. But what they tell us, is that it is unclear. There may be some equations one day that tell us how exactly the universe is. But in this moment we do not know. \ +And maybe the position is just not determined yet, so in the instance of time you measure the position of a particle it gets set up and before that the position is not only unknown to you but also not there. But this is just one ĦInterpretationsĦ2ĦinterpretationĦ of the equations. \ +One small excurse would be the mentioning of the most famous use of the wavefunction or to be more correct the square of the absolute value of it (|Ψ|^2 ): Atomic Orbitals. Every part in the space around the nucleus that has under the given potential a probability of over 90% for finding the electron is part of the orbital. There we can see the changes from just the sheer wavelength to the wavefunction. \ +If we would just have the wavelength it would be enough to say, that the electron is in a specific orbital, but with the wavefunction that just is not the whole question. [\] + diff --git a/assets/0/Bell's Theorem b/assets/0/Bell's Theorem new file mode 100644 index 0000000..3b8538f --- /dev/null +++ b/assets/0/Bell's Theorem @@ -0,0 +1,49 @@ + +[t] One very important gut not too difficult equation in quantum mechanichs is Bell`s Theorem. It disproves tze existence of so called hidden variables(link). \ +It describes tue outcome of an experiment. Luckily the experiment does not really have to be done. The idea is that a pair of entangled photons is emitted by a central source of light. +Special atomic processes can be used as that kind of a light source. The two entangled photons fly away from each other in opposite directions. +In an equal distances there are two polarizers, they are set in different directions. \ +Imagine every photon would habe three hidden variables A, B and C for the three different polarisations. The variable A means that the photon passes through a polarizer which is polarised with the angle ϕ_A . And ϕ_B for B and ϕ_C for C. +If a photon is polarised with an angle of 0° it will get absorbed with a probability of 100% (neglecting tue tunnel effect(link)) by a polarizer with an polarisation-angle of 90°. +The question is how high the probability of absorbing is for an angle different to 90°. +Experiments show that the probability of passing the polarizer with a an angle difference of Δϕ between the direction of polarisation and the angle of the polarizer equals: [\] + +[f] p(Δϕ) = {1}/{2} cos^2 (Δϕ) (1) [\] + +[t] The basis for Bell`s Theorem is that a photon passes through both polarizers A and B. Which is described by p(A, B). If the pair of photons passes through A and B, the probability does not change based on its passing or not passing through C (!C meaning not passing through C): [\] + +[f] p(A, B) = p(A, B, C) + p(A, B, !C) [\] + +[t] If we add the following probabilities to the right hand side of the equation, this side will be bigger equal than the other side then, singe the added probabilities are not 0: [\] + +[f] p(A, B) ≤ p(A, B, C) + p(A, B, !C) + p(A, !B, C) + p(!A, B, !C) [\] + +[t] This can be summarized as: [\] + +[f] p(A, B) ≤ p(A, C) + p(B, !C) [\] + +[t] With: [\] + +[f] p(B, !C) = p(B) - p(B, C) = {1}/{2} - {1}/{2} cos^2 (ϕ_C -ϕ_B ) [\] + +[t] We now make up some values for ϕ_A , ϕ_B and ϕ_C . For example: ϕ_A = 0°, ϕ_B = 30° and ϕ_C = 60°. This then yields with (1): [\] + +[f] {1}/{2} cos^2 (ϕ_B -ϕ_A ) ≤ {1}/{2} cos^2 (ϕ_C -ϕ_A ) + {1}/{2} (1 - cos^2 (ϕ_B -ϕ_C )) [\] + +[f] {1}/{2} cos^2 (30°) ≤ {1}/{2} cos^2 (60°) + {1}/{2} (1 - cos^2 (30°)) [\] + +[f] {3}/{8} ≤ {1}/{8} + {1}/{8} [\] + +[t] This is an obvious contradiction. The conclusion is the non-existence of hidden variables. [\] + + + + + + + + + + + + diff --git a/assets/0/Non Constant Potential b/assets/0/Non Constant Potential new file mode 100644 index 0000000..b8a4a11 --- /dev/null +++ b/assets/0/Non Constant Potential @@ -0,0 +1,47 @@ +[t] For the schroedinger equation with not constant potential we need to solve a rather difficult differential equation (requirements). As a reminder the schroedinger equation in one dimension is: [\] + +[f] - {ħ^2}/{2m}∇^2 ψ(x) + V(x) ψ(x) = E ψ [\] + +[t] Because we know, that the wavefunction is basically the sum of a lot of sines and cosines, which we can due to Euler's formula represent as an exponential function. But to include a more broad range of functions we decided to assume the wavefunction looks like the following: [\] + +[f] ψ(x) = A*e^f(x) * g(x) [\] + +[t] f and g are functions that are not yet defined. We now want to insert our ψ in the schroedinger equation, but for that we have to setermine the second derivative for ψ: [\] + +[f] ψ'(x) = A*e^f(x) *g'(x) + f'(x)*A*e^f(x) *g(x) [\] + +[f] ψ'(x) = A*e^f(x) (g'(x) + f'(x)*g(x)) [\] + +[f] ψ''(x) = f'(x)*A*e^f(x) (g'(x) + f'(x)*g(x)) + A*e^f(x) (g''(x) + f''(x)*g(x) + f'(x)*g'(x)) [\] + +[f] ψ''(x) = A*e^f(x) (2*f'(x) g'(x) + (f'(x))^2 *g(x) + g''(x) + f''(x)*g(x)) [\] + +[f] ψ''(x) = A*e^f(x) g(x) ({2*f'(x) g'(x) + g''(x)}/{g(x)} + (f'(x))^2 + f''(x)) [\] + +[f] ψ''(x) = ψ(x) ({2*f'(x) g'(x) + g''(x)}/{g(x)} + (f'(x))^2 + f''(x)) [\] + +[t] We now insert the second order derivative of ψ(x) in to the schroedinger equation: [\] + +[f] - {ħ^2}/{2m} ψ(x) ({2*f'(x) g'(x) + g''(x)}/{g(x)} + (f'(x))^2 + f''(x)) + V(x) ψ(x) = E ψ(x) [\] + +[f] - {ħ^2}/{2m} ({2*f'(x) g'(x) + g''(x)}/{g(x)} + (f'(x))^2 + f''(x)) + V(x) = E [\] + +[f] {2*f'(x) g'(x) + g''(x)}/{g(x)} + (f'(x))^2 + f''(x) = {-2m}/{ħ^2} (E - V(x)) [\] + +[t] Now you have to find a pair of f(x) and g(x) where that applies to V(x). There usually is an infinte amount of pairs, they represent all different energy levels possible. There are examples in the chapter (link). [\] + + + + + + + + + + + + + + + +