Probability and Quantum Mechanics
Probability is the likelihood that a certain event will occur. Unlike classical physics, quantum physics is not completely deterministic. Probability in the context of quantum mechanics has to do with the likelihood of finding a particle, such as an electron, in a particular region around the nucleus at a particular time.
In 1911 Ernest Rutherford developed a picture explaining the structure of the atom in terms of a positively charged nucleus surrounded by negatively charged electrons in orbits. Later, in 1913, Niels Bohr developed a theory that explained why electrons remain in orbits and do not collapse onto the nucleus. Quantum theory, developed during the 1920s, explained many of the phenomena concerning electrons and their role in atoms. In 1924, using Albert Einstein's special theory of relativity, Louis de Broglie showed that all particles with mass, such as electrons, should have wave-like properties, including a wavelength. Neils Bohr then took this concept and suggested the quantized orbits of atomic electrons where actually electron standing waves around the nucleus. The Bohr atom was a simple two dimensional model. Later, the Schrodinger equation was solved in three dimensions for atomic hydrogen. The cloud-like description comes from the spherical symmetry of these solutions. Therefore, electrons cannot be pictured as localized particles in space but rather should be thought of as clouds of negative charge spread out over the entire orbit. These clouds represent the regions around the nucleus where the probability of finding an electron is the largest. Electrons occupy different energy levels in atoms and have different wave properties.
Because of this idea that electrons behave as particles and waves, it became obvious that a specific type of partial differential equation should be able to describe the position as well as their future behavior. In 1926, Erwin Schrödinger used partial differential equations, Werner Heisenberg's uncertainty principle, and the Hamiltonian function to develop a powerful equation that relates the energy of the electron and the energy of the electric field in which it is situated. Atomic electrons, specifically the hydrogen atom, provided a good laboratory for developing the theory of quantum mechanics. However, quantum mechanics was always intended to be developed generally. The Schrodinger Equation actually derived from a classical mechanics. Schrodinger took a rather standard classical equation of motion and quantized it. That is, he applied the Heisenberg Uncertainty Principle to it turning some of the variables into quantum mechanical operators. The success of the Schrodinger Equation with atomic hydrogen was stunning but by no means the only application. Heisenberg had determined that certain pairs of quantum operators cannot be measured with infinite precision at the same time. For instance, momentum and position cannot both be measured precisely at the same time. The equation Schrödinger derived relates the energy of a system to its wave properties and allows one to predict the energy of the electron and its future behavior, i.e. the probability of finding the electron in a particular region around the nucleus. Shortly after Schrödinger developed his equation, Max Born postulated that the wave function could be used to determine the probability of finding a particle in a particular region at a specific time. This probability is written as |Y(x,t)|2dx = probability at time t of finding the particle at a position x at time t. |Y(x,t)|2 is the observable probability density for finding this particle. If the probability does not change with time then the state is called a stationary state. This does not mean that the particle in a stationary state is not moving but rather that the probability density is stationary.
Because quantum mechanics is basically random in nature, knowing the state does not allow us to predict the result of any measurement with certainty. However, using large numbers of measured instances does allow us to predict the probability of various possible results statistically. Quantum mechanics dver a large region of space as a wave is distributed. Rather that the probability patterns, wave functions, used to describe the electron's motion behave like waves and satisfy wave equations.
Max Born was awarded the Nobel Prize in physics in 1954 for his fundamental research in quantum mechanics, specifically for his statistical interpretation of the wave function. In his collaborations with Wolfgang Pauli and Werner Heisenberg on quantum theory he produced work of fundamental importance in quantum mechanics. His treatment replaced the original quantum theory, which regarded electrons as particles, with a mathematical description representing their observed behavior more accurately. He suggested that the only observable aspect of the wave function was its square, not the wave function itself, and that its square at a given point in space is proportional to the probability of finding the particle at that point in space.
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