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Transcript
b. Proposed interpretation The probability that the particle will be observed to be at a location between x and x+dx at time t 2 2 is given by P( x, t )dx ï½ ï( x, t ) dx , where ï ï½ ï * ï and P(x,t) is the probability density. To be realistic, ï must be continuous and single valued and must be normalized, since ï«ï¥ ï²ï 2 dx ï½ 1 . ïï¥ That is, the particle must be somewhere on the x-axis. This proposed interpretation is to be validated by experiment. c. Solving for the motion The problem, then, is to find ï(x,t), given ï(x,0), rather than x(t) given x(0), as in Newtonian mechanics. We shall have to make the connection between ï 2 and the physical motion of the particle explicit subsequently. That is, later. 3. Particle Experiencing a Conservative Force We draw an analogy with a wave propagating in a dissipative medium. a. Schrödinger equation ï¨2 ï¶ 2ï ï¶ï , ï« U ( x ) ï ï½ i ï¨ 2m ï¶x 2 ï¶t where U(x) is the potential energy function for the external conservative force. The solution of this equation may be easy or difficult, depending on the form of the U(x). Even so, weâre assuming the potential energy function is constant in time. ï b. Time independent Schrödinger equation If the U is not a function of time, then the differential equation is separable, in the usual way. We assume that ïï¨x, t ï© ï½ ï¹ ï¨x ï©ï ï¦ ï¨t ï© . Then ï¨ 2 ï¶ 2ï¹ iï¨ ï¶ï¦ ï« U ï¨x ï© ï½ ï½E, 2 2mï¹ ï¶x ï¦ ï¶t where E is the constant total energy of the particle. The space (x) equation to be solved is ï¨ 2 ï¶ 2ï¹ ï ï« U ( x)ï¹ ï½ Eï¹ . 2m ï¶x 2 To go farther, we need to consider specific potential energy functions. ï 50
