Schrödinger’s+Equation


 * Schrödinger’s Equation **

Schrödinger decided to see if de Broglie’s idea of matter waves could somehow be used to explain Bohr’s model of the atom. Remember that Bohr had suggested that electrons in atoms move around in fixed orbits. (SPDF electron configurations) Schrödinger proposed a new equation that describes not the way a particle moves but the way a wave evolves. He came up with what is now known as a wave equation.

Solving Schrödinger’s equation provides us with a mathematical quantity called the wavefunction. This is where all the probabilistic nature of quantum mechanics comes in. For the case if an electron for instance, the wavefunction does not five us the precise location at a moment in time, only the likelihood of the electron being found somewhere if we were to look for it there. (This is why many textbooks talk about the electron cloud)

It may be helpful to give a rough idea of how a wavefunction changes with time by considering a useful analogy. A burglar has just been released from prison but the local police are convinced he has not turned over a new leaf and can trace his likely whereabouts from the moment he is freed. While they cannot pinpoint his exact location at a given time, they can assign probabilities to burglaries being committed in various places. To begin with, houses close to the prison are at most risk, but the area under threat grows over time. This one-man crime wave spreading through the city can be thought of as a wave of probability. It is not tangible or real, just a set of abstract numbers assigned to parts of the city. In a similar way, a wave function spreads out from the point where the electron was last seen and the wavefunction allows us to assign probabilities.

The police soon find out that a burglary is reported from some address. This alters the probability distribution because they know the thief cannot be far from the crime scene. Likewise, if the electron is detected in a certain location then its wavefunction is instantly altered.

However, there are differences. The police can only assign probabilities to the whereabouts of the burglar because they do not have complete information. After all, the burglar can only be in one place at a given time and has not spread throughout the city. An electron can only be described by its wavefunction when it is not observed. It does not exist as a simple classical particle with a definite location and its influence is spread out through space.