Incoming links: Wave function
A relativistic version of the Schrödinger equation.
Correctly describes spin 0 particles.
The most memorable version of the equation can be written as shown at Section "Klein-Gordon equation in Einstein notation" with Einstein notation and Planck units:
Has some issues which are solved by the Dirac equation:
- it has a second time derivative of the wave function. Therefore, to solve it we must specify not only the initial value of the wave equation, but also the derivative of the wave equation,As mentioned at Advanced quantum mechanics by Freeman Dyson (1951) and further clarified at: physics.stackexchange.com/questions/340023/cant-the-negative-probabilities-of-klein-gordon-equation-be-avoided, this would lead to negative probabilities.
- the modulus of the wave function is not constant and therefore not always one, and therefore cannot be interpreted as a probability density anymore
- since we are working with the square of the energy, we have both positive and negative value solutions. This is also a features of the Dirac equation however.
Bibliography:
- Video "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=600
- An Introduction to QED and QCD by Jeff Forshaw (1997) 1.2 "Relativistic Wave Equations" and 1.4 "The Klein Gordon Equation" gives some key ideas
- 2011 PHYS 485 lecture videos by Roger Moore from the University of Alberta at around 7:30
- www.youtube.com/watch?v=WqoIW85xwoU&list=PL54DF0652B30D99A4&index=65 "L2. The Klein-Gordon Equation" by doctorphys
- sites.ualberta.ca/~gingrich/courses/phys512/node21.html from Advanced quantum mechanics II by Douglas Gingrich (2004)
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:
- 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 Schrödinger equation for a free one dimensional particle.
Then, with that in mind, the general form of the Schrödinger equation is:where:
Equation 1.
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 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.
Is the only atom that has a closed form solution, which allows for very good predictions, and gives awesome intuition about the orbitals in general.
It is arguably the most important solution of the Schrodinger equation.
In particular, it predicts:
- the major spectral line of the hydrogen atom by taking the difference between energy levels
The explicit solution can be written in terms of spherical harmonics.
A Better Way To Picture Atoms by minutephysics (2021)
Source. Renderings based on the exact Schrödinger equation solution for the hydrogen atom that depict wave function concentration by concentration of small balls, and angular momentum by how fast the balls rotate at each point. Mentions that the approach is inspired by de Broglie-Bohm theory.As mentioned at Section "Plancherel theorem", some people call this part of Plancherel theorem, while others say it is just a corollary.
This is an important fact in quantum mechanics, since it is because of this that it makes sense to talk about position and momentum space as two dual representations of the wave function that contain the exact same amount of information.
The wave equation contains the entire state of a particle.
From mathematical formulation of quantum mechanics remember that the wave equation is a vector in Hilbert space.
And a single vector can be represented in many different ways in different basis, and two of those ways happen to be the position and the momentum representations.
More importantly, position and momentum are first and foremost operators associated with observables: the position operator and the momentum operator. And both of their eigenvalue sets form a basis of the Hilbert space according to the spectral theorem.
When you represent a wave equation as a function, you have to say what the variable of the function means. And depending on weather you say "it means position" or "it means momentum", the position and momentum operators will be written differently.
This is well shown at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)".
Furthermore, the position and momentum representations are equivalent: one is the Fourier transform of the other: position and momentum space. Remember that notably we can always take the Fourier transform of a function in due to Carleson's theorem.
Then the uncertainty principle follows immediately from a general property of the Fourier transform: en.wikipedia.org/w/index.php?title=Fourier_transform&oldid=961707157#Uncertainty_principle
In precise terms, the uncertainty principle talks about the standard deviation of two measures.
We can visualize the uncertainty principle more intuitively by thinking of a wave function that is a real flat top bump function with a flat top in 1D. We can then change the width of the support, but when we do that, the top goes higher to keep probability equal to 1. The momentum is 0 everywhere, except in the edges of the support. Then:
- to localize the wave in space at position 0 to reduce the space uncertainty, we have to reduce the support. However, doing so makes the momentum variation on the edges more and more important, as the slope will go up and down faster (higher top, and less x space for descent), leading to a larger variance (note that average momentum is still 0, due to to symmetry of the bump function)
- to localize the momentum as much as possible at 0, we can make the support wider and wider. This makes the bumps at the edges smaller and smaller. However, this also obviously delocalises the wave function more and more, increasing the variance of x
Bibliography:
- www.youtube.com/watch?v=bIIjIZBKgtI&list=PL54DF0652B30D99A4&index=59 "K2. Heisenberg Uncertainty Relation" by doctorphys (2011)
- physics.stackexchange.com/questions/132111/uncertainty-principle-intuition Uncertainty Principle Intuition on Physics Stack Exchange