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The discussion at this point may seem somewhat abstract, but the physical implications of all of this are profound and wide ranging. We will see many examples of eigen value equations for real properties of dynamic systems in later chapters. 4. 3 Equations of motion (Postulate 3) The third postulate states that: All state functions satisfy the ‘‘time-dependent Schro¨dinger equation’’: i"h @ ^ Y; Y¼H @t (2:16) ^ is the Hamiltonian of the system. The Hamiltonian is the operator correspondwhere H ing to the total energy of the system.
This is what quantum mechanics says. How can one understand it intuitively? Superficially, it is not as unreasonable as one might think on first sight. As we recall, in quantum mechanics, the result of measurements must always be understood in the probability sense. ’’ Clearly, if this is the only information about the state of the particle, the particle in the ensemble can be anywhere with equal probability. It must be emphasized, however, that Heisenberg’s uncertainty principle goes far deeper than this intuitive argument about semantics.
Postulate 1 tells us that the state of any system is completely specified by the state function. Thus, solution of Schro¨dinger’s equation for Yð~ r; tÞ describes completely the state of the dynamic system at all times once the initial condition Yð~ r; t ¼ 0Þ and the Hamiltonian are known. 3 Equations of motion (Postulate 3) 19 ^ r; tÞ. In the case when it is The Hamiltonian in general can be a function of time, Hð~ not a function of time (the system is ‘‘conservative’’ or the potential energy of the ^ rÞ is a function of the position coordinates only), the time-dependent system Vð~ Schro¨dinger equation is: i" h !