A single electron of mass m can move freely along a one-dimensionl gold nanowire. Let x be the position coordinate of the electron along the wire. (a) Let ø (x) be the wave function of the electron. The quantity |ø (x)|² has dimensions of inverse length. Explain very briefly the meaning of this quantity as a probability density. (b) Let us assume that $ (x) = Asin (3kox) (2) where A and ko are fixed, positive constants. Establish whether this wave function represents an eigenstate of momentum p. Justify your answer. Hint: the momentum operator is p = -ih. (c) Establish whether the wave function (x) given in Eq. represents an eigenstate of kinetic energy K. Justify your answer. Hint: the kinetic energy operator is K = p²/2m.
A single electron of mass m can move freely along a one-dimensionl gold nanowire. Let x be the position coordinate of the electron along the wire. (a) Let ø (x) be the wave function of the electron. The quantity |ø (x)|² has dimensions of inverse length. Explain very briefly the meaning of this quantity as a probability density. (b) Let us assume that $ (x) = Asin (3kox) (2) where A and ko are fixed, positive constants. Establish whether this wave function represents an eigenstate of momentum p. Justify your answer. Hint: the momentum operator is p = -ih. (c) Establish whether the wave function (x) given in Eq. represents an eigenstate of kinetic energy K. Justify your answer. Hint: the kinetic energy operator is K = p²/2m.
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Transcribed Image Text:A single electron of mass m can move freely along a one-dimensionl gold
nanowire. Let x be the position coordinate of the electron along the wire.
(a) Let ø (x) be the wave function of the electron. The quantity |ø (x)|
has dimensions of inverse length. Explain very briefly the meaning
of this quantity as a probability density.
(b) Let us assume that
$ (x) = A sin (3kox)
(2)
where A and ko are fixed, positive constants. Establish whether this
wave function represents an eigenstate of momentum p. Justify your
answer.
Hint: the momentum operator is p
-ih.
-
(c) Establish whether the wave function (x) given in Eq. (2) represents
an eigenstate of kinetic energy K. Justify your answer.
Hint: the kinetic energy operator is K = p²/2m.
(d) Let us now assume that the gold nanowire mentioned above is not
infinite, but extends over a finite length from r= 0 to x = L. Inside
this region, the potential energy of the electron is zero, but outside
this region the potential energy is infinite (particle-in-a-box). Write
the boundary conditions that the wave function (x) must obey at
x = 0 and x = L, explaining their origin.
(e) Use the boundary conditions you have obtained to deduce a formula
giving the allowed values of the parameter ko in Eq. (2). Express
your results as a function of L.
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