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A particle is in the ground state of an infinite square well potential given by,
0 for -a sxsa
V(x) =
otherwise
The probability to find the particle in the interval between
a
and
2
is

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(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel through the potential barrier of the width L, the difference between the barrier height U and the incident energy E of the particle with mass m has to be close. Using the transmission probability given in the text book / lecture, obtain the energy difference U-E which gives the transmission probability of exp(-2). (ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as "shadows" of each other. Energy U ---E« Electron X L L (iii) We now consider a 3-dimensional infinite square well potential having the length of the x, y, and z directions to be all L. V=L**3 is the volume of the cube of this potential. We consider energy level of a single particle (boson)…
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