A quantum mechanical particle is confined to a one-dimensional infinite potential well described by the function V(x) = 0 in the region 0 < x < L, V(x) = ∞ elsewhere. The normalised eigenfunctions for a particle moving in this potential are: Yn(x) = = NTT sin -X Vī L where n = 1, 2, 3, ... a) Write down the expression for the corresponding probability density function. Sketch the shape of this function for a particle in the ground state (n = 1).
A quantum mechanical particle is confined to a one-dimensional infinite potential well described by the function V(x) = 0 in the region 0 < x < L, V(x) = ∞ elsewhere. The normalised eigenfunctions for a particle moving in this potential are: Yn(x) = = NTT sin -X Vī L where n = 1, 2, 3, ... a) Write down the expression for the corresponding probability density function. Sketch the shape of this function for a particle in the ground state (n = 1).
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
Transcribed Image Text:A quantum mechanical particle is confined to a one-dimensional infinite potential well
described by the function V(x) = 0 in the region 0 < x < L, V(x) = ∞ elsewhere.
The normalised eigenfunctions for a particle moving in this potential are:
Yn(x)
=
√
2
Nπ
sin -X
L L
where n = 1, 2, 3, ..
a) Write down the expression for the corresponding probability density function. Sketch
the shape of this function for a particle in the ground state (n = 1).
b) Annotate your sketch to show the probability density function for a classical particle
moving at constant speed in the well. Give a short justification for the shape of your
sketch.
c) Briefly describe, with the aid of a sketch or otherwise, the way in which the quantum
and the classical probability density functions are consistent with the correspondence
principle for large values of n.
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