Computing the time dependent amplitude function
K
K
K
Now that we have A(q ) , how do we find the time dependence of Ψ ( x, t ) ? Since A(q ) is
the coefficient vector for the eigenfunctions of H, we know the time dependence is
accounted for by multiplying each coefficient by e −iEt = . So, we now create a vector E,
indexed by q.
⎛ q12 ⎞
⎜ 2⎟
q
K = 2 JJG2 = 2 ⎜ 22 ⎟
⎜q ⎟
⋅q =
E=
2m
2m ⎜ 3 ⎟
⎜ # ⎟
⎜ q2 ⎟
⎝ m⎠
Now, if we take the exponential of −it = times each element, where t is the time we wish
to evaluateΨ ( x ) at, we get
⎛ e −iE1t = ⎞
⎜ − iE t = ⎟
K ⎜e 2 ⎟
ς =
⎜ # ⎟
⎜⎜ −iEmt = ⎟⎟
⎝e
⎠
Now, if weK perform element by element multiplication (MATLAB® command is “.*”) on
K
ς (q ) and A(q ) we get:
⎛ A(q1 )ς (q1 ) ⎞ ⎛ A(q1 )e− iE1t = ⎞
⎜
⎟ ⎜
− iE2t = ⎟
K
K
ς
A
(
q
)
(
q
)
A
(
q
)
e
K
2
2
2
⎟
⎟=⎜
At (q) = A(q) ⋅ *ς (q) = ⎜
⎜
⎟
⎜
⎟
#
#
⎟
⎜
⎟ ⎜⎜
− iEm t = ⎟
ς
A
(
q
)
(
q
)
A
(
q
)
e
m
m ⎠
⎝
m
⎝
⎠
Now, we have taken account for the time dependence by modifying our amplitude
function (note the subscript t to denote that this is A(q ) at a particular time t). The last
chore is to now compute the wave function in x-space from the modified amplitude
function.