Particle in Step Potential With Energy E V

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C. Particle in Step Potential With Energy E V

o In Figure 10.2 is described particle in step potential with energy E V o . Figure 10.2. Particle in Step Potential With Energy E V o According to quantum mechanics, in area x0 with Vx=0 full filled time independent of Schrodinger equation: 2 2 2 2 x E dx x d m      It has general solution: x ik x ik Be Ae x 1 1     where mE k 2 1 1   in area x0 with Vx= V o full filled: 2 2 2 2 x E x V dx x d m o        2 2 2 2 x V E dx x d m o       10.16 Equation 10.16 has general solution: x ik x ik De Ce x 2 2     10.17 where 2 1 2 o V E m k    Vx V o Vx=0 x E 85 Then in ψx is applied boundary condition in order wave function has well behave in all interval of x. 1. Particle come from left side to the right side of step potential at x=0 may be undergoes ressistance, hence reflection is happen. But in x0 there is no wave which propagate in left direction, hence coefficient D in equation 10.17 equals to zero D=0. 2. Condition of well behave function is ψx continue at x=0, it is obtained: A + B = C 10.18 3. Condition of well behave function ψ’x also continue at x=0: In area x0 it is obtained: x ik x ik Be ik Ae ik x 1 1 1 1     In area x0 it is obtained: x ik Ce ik x 2 2   Continuity condition at x=0 produce:   C k B A k C ik B ik A ik 2 1 2 1 1     10.19 Based on equation 10.18 and 10.19, it can be determined coefficient B and C which are stated in A as follow: B – C = -A A k C k B k 1 2 1   The equation can be written: A k C k B k 2 2 2    A k C k B k 1 2 1   Sum of both equation produce: A k k B k k 2 1 2 1        A k k k k B 2 1 2 1    10.20 Furthermore coefficient C can be determined:           2 1 2 1 2 1 2 1 2 1 k k A k k A k k C A k k k k A C B A C            86   A k k k C 2 1 1 2   10.21 The eigen function which determined are: 1. In area x0: x ik x ik e k k k k A Ae x 1 1 2 1 2 1           2. In area x0: x ik e k k k A x 2 2 1 1 2        Refflection Coefficient R:         A k k A k k A k k A k k A A B B v v R 2 1 2 1 2 1 2 1 1 1                 2 2 1 2 2 1 k k k k R    10.22 where v 1 is velocity of particle which has the same value with group velocity Transmission Coefficient T: A A k k k A A k k k k k A A C C v v T                  2 1 1 2 1 1 1 2 1 2 2 2   2 2 1 2 1 4 k k k k T   10.23 Sum of R and T produce:           2 2 1 2 2 2 1 2 1 2 2 1 2 1 2 2 2 1 2 1 2 2 1 2 1 2 2 1 2 2 1 2 4 2 4 k k k k k k T R k k k k k k k k T R k k k k k k k k T R                  87     1 2 2 1 2 2 1      k k k k T R 10.24 Equation 10.24 shows the conservation of probability.

D. Particle in Barrier Potential with EV