This describes a wave propagating in the

# This describes a wave propagating in the - (3.9 Using this...

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This describes a wave propagating in the direction with wavelength , and frequency . We interpret this plane wave as a propagating beam of particles. If we define the probability as the square of the wave function, it is not very sensible to take the real part of the exponential: the probability would be an oscillating function of for given . If we take the complex function , however, the probability, defined as the absolute value squared, is a constant ( ) independent of and , which is very sensible for a beam of particles. Thus we conclude that the wave function is complex, and the probability density is . Using de Broglie's relation (3.5) we find (3.6) The other of de Broglie's relations can be used to give (3.7) One of the important goals of quantum mechanics is to generalise classical mechanics. We shall attempt to generalise the relation between momenta and energy, (3.8) to the quantum realm. Notice that

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Unformatted text preview: (3.9) Using this we can guess a wave equation of the form (3.10) Actually using the definition of energy when the problem includes a potential, (3.11) (when expressed in momenta, this quantity is usually called a "Hamiltonian") we find the time-dependent Schrödinger equation (3.12) We shall only spend limited time on this equation. Initially we are interested in the time-independent Schrödinger equation, where the probability is independent of time. In order to reach this simplification, we find that must have the form (3.13) If we substitute this in the time-dependent equation, we get (using the product rule for differentiation) (3.14) Taking away the common factor we have an equation for that no longer contains time, the time-indepndent Schrödinger equation (3.15) The corresponding solution to the time-dependent equation is the standing wave ( 3.13 )....
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This describes a wave propagating in the - (3.9 Using this...

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