Time-dependent Schrodinger equation Notes | EduRev

: Time-dependent Schrodinger equation Notes | EduRev

 Page 1


6. Time Evolution in Quantum Mechanics 
6.1 Time-dependent Schr¨ odinger equation 
6.1.1 Solutions to the Schr¨ odinger equation 
6.1.2 Unitary Evolution 
6.2 Evolution of wave-packets 
6.3 Evolution of operators and expectation values 
6.3.1 Heisenberg Equation 
6.3.2 Ehrenfest’s theorem 
6.4 Fermi’s Golden Rule 
Until now we used quantum mechanics to predict properties of atoms and nuclei. Since we were interested mostly 
in the equilibrium states of nuclei and in their energies, we only needed to look at a time-independent description 
of quantum-mechanical systems. To describe dynamical processes, such as radiation decays, scattering and nuclear 
reactions, we need to study how quantum mechanical systems evolve in time. 
6.1 Time-dependent Schr o ¨dinger equation 
When we ?rst introduced quantum mechanics, we saw that the fourth postulate of QM states that:
 
The evolution of a closed system is unitary (reversible). The evolution is given by the time-dependent Schr ¨
 odinger 
equation 
?|?)
iI = H|?)
?t 
where H is the Hamiltonian of the system (the energy operator) and I is the reduced Planck constant 
(I = h/2p with h the Planck constant, allowing conversion from energy to frequency units). 
We will focus mainly on the Schr¨ odinger equation to describe the evolution of a quantum-mechanical system. The 
statement that the evolution of a closed quantum system is unitary is however more general. It means that the state 
of a system at a later time t is given by |?(t)) = U(t)|?(0)), where U(t) is a unitary operator. An operator is unitary 
if its adjoint U
† 
(obtained by taking the transpose and the complex conjugate of the operator, U
† 
= (U 
*
)
T 
) is equal 
U
-1
to its inverse: U
† 
= or UU
† 
= 1 1. 
Note that the expression |?(t)) = U(t)|?(0)) is an integral equation relating the state at time zero with the state at 
time t. For example, classically we could write that x(t) = x(0)+ vt (where v is the speed, for constant speed). We 
can as well write a di?erential equation that provides the same information: the Schr¨ odinger equation. Classically 
for example, (in the example above) the equivalent di?erential equation would be 
dx 
= v (more generally we would 
dt 
have Newton’s equation linking the acceleration to the force). In QM we have a di?erential equation that control the 
evolution of closed systems. This is the Schr¨ odinger equation: 
??(x, t)
iI = H?(x, t)
?t 
where H is the system’s Hamiltonian. The solution to this partial di?erential equation gives the wavefunction ?(x, t) 
at any later time, when ?(x, 0) is known. 
6.1.1 Solutions to the Schr o ¨dinger equation 
2 
p ˆ
We ?rst try to ?nd a solution in the case where the Hamiltonian H = +V (x, t) is such that the potential V (x, t)
2m 
is time independent (we can then write V (x)). In this case we can use separation of variables to look for solutions. 
That is, we look for solutions that are a product of a function of position only and a function of time only: 
?(x, t) = ?(x)f(t) 
83
Page 2


6. Time Evolution in Quantum Mechanics 
6.1 Time-dependent Schr¨ odinger equation 
6.1.1 Solutions to the Schr¨ odinger equation 
6.1.2 Unitary Evolution 
6.2 Evolution of wave-packets 
6.3 Evolution of operators and expectation values 
6.3.1 Heisenberg Equation 
6.3.2 Ehrenfest’s theorem 
6.4 Fermi’s Golden Rule 
Until now we used quantum mechanics to predict properties of atoms and nuclei. Since we were interested mostly 
in the equilibrium states of nuclei and in their energies, we only needed to look at a time-independent description 
of quantum-mechanical systems. To describe dynamical processes, such as radiation decays, scattering and nuclear 
reactions, we need to study how quantum mechanical systems evolve in time. 
6.1 Time-dependent Schr o ¨dinger equation 
When we ?rst introduced quantum mechanics, we saw that the fourth postulate of QM states that:
 
The evolution of a closed system is unitary (reversible). The evolution is given by the time-dependent Schr ¨
 odinger 
equation 
?|?)
iI = H|?)
?t 
where H is the Hamiltonian of the system (the energy operator) and I is the reduced Planck constant 
(I = h/2p with h the Planck constant, allowing conversion from energy to frequency units). 
We will focus mainly on the Schr¨ odinger equation to describe the evolution of a quantum-mechanical system. The 
statement that the evolution of a closed quantum system is unitary is however more general. It means that the state 
of a system at a later time t is given by |?(t)) = U(t)|?(0)), where U(t) is a unitary operator. An operator is unitary 
if its adjoint U
† 
(obtained by taking the transpose and the complex conjugate of the operator, U
† 
= (U 
*
)
T 
) is equal 
U
-1
to its inverse: U
† 
= or UU
† 
= 1 1. 
Note that the expression |?(t)) = U(t)|?(0)) is an integral equation relating the state at time zero with the state at 
time t. For example, classically we could write that x(t) = x(0)+ vt (where v is the speed, for constant speed). We 
can as well write a di?erential equation that provides the same information: the Schr¨ odinger equation. Classically 
for example, (in the example above) the equivalent di?erential equation would be 
dx 
= v (more generally we would 
dt 
have Newton’s equation linking the acceleration to the force). In QM we have a di?erential equation that control the 
evolution of closed systems. This is the Schr¨ odinger equation: 
??(x, t)
iI = H?(x, t)
?t 
where H is the system’s Hamiltonian. The solution to this partial di?erential equation gives the wavefunction ?(x, t) 
at any later time, when ?(x, 0) is known. 
6.1.1 Solutions to the Schr o ¨dinger equation 
2 
p ˆ
We ?rst try to ?nd a solution in the case where the Hamiltonian H = +V (x, t) is such that the potential V (x, t)
2m 
is time independent (we can then write V (x)). In this case we can use separation of variables to look for solutions. 
That is, we look for solutions that are a product of a function of position only and a function of time only: 
?(x, t) = ?(x)f(t) 
83
Then, when we take the partial derivatives we have that 
??(x, t) df(t) ??(x, t) d?(x) ?
2
?(x, t) d
2
?(x) 
= ?(x), = f(t) and = f(t)
?t dt ?x dx ?x
2 
dx
2 
The Schr¨ odinger equation simpli?es to 
df(t) I
2 
d
2
?(x)
iI ?(x) = - f(t)+V (x)?(x)f(t)
dt 2m x
2 
Dividing by ?(x, t) we have: 
df(t) 1 I
2 
d
2
?(x) 1 
iI = - + V (x)
dt f(t) 2m x
2 
?(x) 
Now the LHS is a function of time only, while the RHS is a function of position only. For the equation to hold, both 
sides have then to be equal to a constant (separation constant): 
df(t) 1 I
2 
d
2
?(x) 1 
iI = E, - + V (x) = E 
dt f(t) 2m x
2 
?(x) 
The two equations we ?nd are a simple equation in the time variable: 
-i 
Et df(t) i 
i
= - Ef(t), ? f(t) = f(0)e 
dt I 
and 
I
2 
d
2
?(x) 1 
- + V (x) = E 
2m x
2 
?(x) 
that we have already seen as the time-independent Schr¨ odinger equation. We have extensively studied the solutions 
of the this last equation, as they are the eigenfunctions of the energy-eigenvalue problem, giving the stationary (equi­
librium) states of quantum systems. Note that for these stationary solutions ?(x) we can still ?nd the corresponding 
total wavefunction, given as stated above by ?(x, t) = ?(x)f(t), which does describe also the time evolution of the 
system: 
?(x, t) = ?(x)e 
-i 
Et
i 
Does this mean that the states that up to now we called stationary are instead evolving in time? 
The answerisyes,but witha caveat. Althoughthestatesthemselves evolve asstatedabove, any measurablequantity 
(such as the probability density |?(x, t)|
2 
or the expectation values of observable, (A) = 
J 
?(x, t)
*
A[?(x, t)]) are still 
time-independent. (Check it!) 
Thus we were correct in calling these states stationary and neglecting in practice their time-evolution when studying 
the properties of systems they describe. 
Notice that the wavefunction built from one energy eigenfunction, ?(x, t) = ?(x)f(t), is only a particular solution 
of the Schr¨ odinger equation, but many other are possible. These will be complicated functions of space and time, 
whose shape will depend on the particular form of the potential V (x). How can we describe these general solutions? 
We know that in general we can write a basis given by the eigenfunction of the Hamiltonian. These are the functions 
{?(x)} (as de?ned above by the time-independent Schr¨ odinger equation). The eigenstate of the Hamiltonian do not 
evolve. However we can write any wavefunction as 
L
?(x, t) = c
k
(t)?
k
(x) 
k 
This just corresponds to express the wavefunction in the basis given by the energy eigenfunctions. As usual, the 
coe?cients c
k
(t) can be obtained at any instant in time by taking the inner product: (?
k
|?(x, t)). 
What is the evolution of such a function? Substituting in the Schr¨ odinger equation we have 
?(
L
k 
c
k
(t)?
k
(x)) 
L
iI = c
k
(t)H?
k
(x)
?t 
k 
that becomes 
L 
?(c
k
(t)) 
L 
iI ?
k
(x) = c
k
(t)E
k
?
k
(x)
?t 
k k 
84
Page 3


6. Time Evolution in Quantum Mechanics 
6.1 Time-dependent Schr¨ odinger equation 
6.1.1 Solutions to the Schr¨ odinger equation 
6.1.2 Unitary Evolution 
6.2 Evolution of wave-packets 
6.3 Evolution of operators and expectation values 
6.3.1 Heisenberg Equation 
6.3.2 Ehrenfest’s theorem 
6.4 Fermi’s Golden Rule 
Until now we used quantum mechanics to predict properties of atoms and nuclei. Since we were interested mostly 
in the equilibrium states of nuclei and in their energies, we only needed to look at a time-independent description 
of quantum-mechanical systems. To describe dynamical processes, such as radiation decays, scattering and nuclear 
reactions, we need to study how quantum mechanical systems evolve in time. 
6.1 Time-dependent Schr o ¨dinger equation 
When we ?rst introduced quantum mechanics, we saw that the fourth postulate of QM states that:
 
The evolution of a closed system is unitary (reversible). The evolution is given by the time-dependent Schr ¨
 odinger 
equation 
?|?)
iI = H|?)
?t 
where H is the Hamiltonian of the system (the energy operator) and I is the reduced Planck constant 
(I = h/2p with h the Planck constant, allowing conversion from energy to frequency units). 
We will focus mainly on the Schr¨ odinger equation to describe the evolution of a quantum-mechanical system. The 
statement that the evolution of a closed quantum system is unitary is however more general. It means that the state 
of a system at a later time t is given by |?(t)) = U(t)|?(0)), where U(t) is a unitary operator. An operator is unitary 
if its adjoint U
† 
(obtained by taking the transpose and the complex conjugate of the operator, U
† 
= (U 
*
)
T 
) is equal 
U
-1
to its inverse: U
† 
= or UU
† 
= 1 1. 
Note that the expression |?(t)) = U(t)|?(0)) is an integral equation relating the state at time zero with the state at 
time t. For example, classically we could write that x(t) = x(0)+ vt (where v is the speed, for constant speed). We 
can as well write a di?erential equation that provides the same information: the Schr¨ odinger equation. Classically 
for example, (in the example above) the equivalent di?erential equation would be 
dx 
= v (more generally we would 
dt 
have Newton’s equation linking the acceleration to the force). In QM we have a di?erential equation that control the 
evolution of closed systems. This is the Schr¨ odinger equation: 
??(x, t)
iI = H?(x, t)
?t 
where H is the system’s Hamiltonian. The solution to this partial di?erential equation gives the wavefunction ?(x, t) 
at any later time, when ?(x, 0) is known. 
6.1.1 Solutions to the Schr o ¨dinger equation 
2 
p ˆ
We ?rst try to ?nd a solution in the case where the Hamiltonian H = +V (x, t) is such that the potential V (x, t)
2m 
is time independent (we can then write V (x)). In this case we can use separation of variables to look for solutions. 
That is, we look for solutions that are a product of a function of position only and a function of time only: 
?(x, t) = ?(x)f(t) 
83
Then, when we take the partial derivatives we have that 
??(x, t) df(t) ??(x, t) d?(x) ?
2
?(x, t) d
2
?(x) 
= ?(x), = f(t) and = f(t)
?t dt ?x dx ?x
2 
dx
2 
The Schr¨ odinger equation simpli?es to 
df(t) I
2 
d
2
?(x)
iI ?(x) = - f(t)+V (x)?(x)f(t)
dt 2m x
2 
Dividing by ?(x, t) we have: 
df(t) 1 I
2 
d
2
?(x) 1 
iI = - + V (x)
dt f(t) 2m x
2 
?(x) 
Now the LHS is a function of time only, while the RHS is a function of position only. For the equation to hold, both 
sides have then to be equal to a constant (separation constant): 
df(t) 1 I
2 
d
2
?(x) 1 
iI = E, - + V (x) = E 
dt f(t) 2m x
2 
?(x) 
The two equations we ?nd are a simple equation in the time variable: 
-i 
Et df(t) i 
i
= - Ef(t), ? f(t) = f(0)e 
dt I 
and 
I
2 
d
2
?(x) 1 
- + V (x) = E 
2m x
2 
?(x) 
that we have already seen as the time-independent Schr¨ odinger equation. We have extensively studied the solutions 
of the this last equation, as they are the eigenfunctions of the energy-eigenvalue problem, giving the stationary (equi­
librium) states of quantum systems. Note that for these stationary solutions ?(x) we can still ?nd the corresponding 
total wavefunction, given as stated above by ?(x, t) = ?(x)f(t), which does describe also the time evolution of the 
system: 
?(x, t) = ?(x)e 
-i 
Et
i 
Does this mean that the states that up to now we called stationary are instead evolving in time? 
The answerisyes,but witha caveat. Althoughthestatesthemselves evolve asstatedabove, any measurablequantity 
(such as the probability density |?(x, t)|
2 
or the expectation values of observable, (A) = 
J 
?(x, t)
*
A[?(x, t)]) are still 
time-independent. (Check it!) 
Thus we were correct in calling these states stationary and neglecting in practice their time-evolution when studying 
the properties of systems they describe. 
Notice that the wavefunction built from one energy eigenfunction, ?(x, t) = ?(x)f(t), is only a particular solution 
of the Schr¨ odinger equation, but many other are possible. These will be complicated functions of space and time, 
whose shape will depend on the particular form of the potential V (x). How can we describe these general solutions? 
We know that in general we can write a basis given by the eigenfunction of the Hamiltonian. These are the functions 
{?(x)} (as de?ned above by the time-independent Schr¨ odinger equation). The eigenstate of the Hamiltonian do not 
evolve. However we can write any wavefunction as 
L
?(x, t) = c
k
(t)?
k
(x) 
k 
This just corresponds to express the wavefunction in the basis given by the energy eigenfunctions. As usual, the 
coe?cients c
k
(t) can be obtained at any instant in time by taking the inner product: (?
k
|?(x, t)). 
What is the evolution of such a function? Substituting in the Schr¨ odinger equation we have 
?(
L
k 
c
k
(t)?
k
(x)) 
L
iI = c
k
(t)H?
k
(x)
?t 
k 
that becomes 
L 
?(c
k
(t)) 
L 
iI ?
k
(x) = c
k
(t)E
k
?
k
(x)
?t 
k k 
84
For each ?
k 
we then have the equation in the coe?cients only 
dc
k 
-i 
E
k
t 
i
iI = E
k
c
k
(t) ? c
k
(t) = c
k
(0)e 
dt 
A general solution of the Schr¨ odinger equation is then 
E
k
t
L 
-i 
i
?(x, t) = c
k
(0)e ?
k
(x) 
k 
Obs. We can de?ne the eigen-frequencies I?
k 
= E
k 
from the eigen-energies. Thus we see that the wavefunction is a 
superposition of waves ?
k 
propagating in time each with a di?erent frequency ?
k
. 
The behavior of quantum systems –even particles– thus often is similar to the propagation of waves. One example 
is the di?raction pattern for electrons (and even heavier objects) when scattering from a slit. We saw an example in 
the electron di?raction video at the beginning of the class. 
Obs. What is the probability of measuring a certain energy E
k 
at a time t? It is given by the coe?cient of the ?
k 
-i 
i
eigenfunction, |c
k
(t)|
2 
= |c
k
(0)e
E
k
t 
|
2 
= |c
k
(0)|
2
. This means that the probability for the given energy is constant, 
does not change in time. Energy is then a so-called constant of the motion. This is true only for the energy eigenvalues,
 
not for other observables‘.
 
Example: Considerinsteadtheprobabilityof?ndingthesystemata certainposition, p(x) = |?(x, t)|
2
. This of course
 
changes in time. For example, let ?(x, 0) = c
1
(0)?
1
(x)+c
2
(0)?
2
(x), with |c
1
(0)|
2 
+ |c
2
(0)|
2 
= |c
1
|
2 
+ |c
2
|
2 
= 1 (and
 
?
1,2 
normalized energy eigenfunctions. Then at a later time we have ?(x, 0) = c
1
(0)e
-i?1t
?
1
(x)+c
2
(0)e
-i?2t
?
2
(x).
 
What is p(x, t)?

 
-i?2 t
?
2
(x)
 
2 
 
c
1
(0)e 
-i?1t
?
1
(x)+c
2
(0)e 
 
* -i(?2-?1)t * i(?2-?1)t 
= |c
1
(0)|
2
|?
1
(x)|
2 
+ |c
2
(0)|
2
|?
2
(x)|
2 
+ c
1
c
2
?
1
* 
?
2
e + c
1
c
2
?
1
? 
* 
2
e 
[ ]
* -i(?2 -?1)t
= |c
1
|
2 
+ |c
2
|
2 
+ 2Rec
1
c
2
? 
* 
1
?
2
e 
The last term describes a wave interference between di?erent components of the initial wavefunction.
 
Obs.: The expressions found above for the time-dependent wavefunction are only valid if the potential is itself
 
time-independent. If this is not the case, the solutions are even more di?cult to obtain.
 
6.1.2 Unitary Evolution 
Wesawtwoequivalentformulationofthequantum mechanicalevolution,theSchr¨ odingerequationandthe Heisenberg 
equation. We now present a third possible formulation: following the 4
th 
postulate we express the evolution of a state 
in terms of a unitary operator, called the propagator: 
ˆ
?(x, t) = U(t)?(x, 0) 
with U
ˆ
†
U
ˆ
= 1 1. (Notice that a priori the unitary operator U
ˆ
could also be a function of space). We can show that 
this is equivalent to the Schr¨ odinger equation, by verifying that ?(x, t) above is a solution: 
? 
ˆ
? 
ˆ
U?(x, 0) U 
H 
ˆ
iI = U?(x, 0) ? iI = HU
ˆ
?t ?t 
where in the second step we used the fact that since the equation holds for any wavefunction ? it must hold for the 
operator themselves. If the Hamiltonian is time independent, the second equation can be solved easily, obtaining: 
?U
ˆ
-iHt/n
iI = HU
ˆ
? U
ˆ
(t) = e 
?t 
iHt/n -iHt/n
where we set U
ˆ
(t = 0) = 1 1. Notice that as desired U
ˆ
is unitary, U
ˆ
†
U
ˆ
= e e = 1 1. 
6.2 Evolution of wave-packets 
In Section 6.1.1 we looked at the evolution of a general wavefunction under a time-independent Hamiltonian. The 
solution to the Schr¨ odinger equation was given in terms of a linear superposition of energy eigenfunctions, each 
acquiring a time-dependent phase factor. The solution was then the superposition of waves each with a di?erent 
frequency. 
85
Page 4


6. Time Evolution in Quantum Mechanics 
6.1 Time-dependent Schr¨ odinger equation 
6.1.1 Solutions to the Schr¨ odinger equation 
6.1.2 Unitary Evolution 
6.2 Evolution of wave-packets 
6.3 Evolution of operators and expectation values 
6.3.1 Heisenberg Equation 
6.3.2 Ehrenfest’s theorem 
6.4 Fermi’s Golden Rule 
Until now we used quantum mechanics to predict properties of atoms and nuclei. Since we were interested mostly 
in the equilibrium states of nuclei and in their energies, we only needed to look at a time-independent description 
of quantum-mechanical systems. To describe dynamical processes, such as radiation decays, scattering and nuclear 
reactions, we need to study how quantum mechanical systems evolve in time. 
6.1 Time-dependent Schr o ¨dinger equation 
When we ?rst introduced quantum mechanics, we saw that the fourth postulate of QM states that:
 
The evolution of a closed system is unitary (reversible). The evolution is given by the time-dependent Schr ¨
 odinger 
equation 
?|?)
iI = H|?)
?t 
where H is the Hamiltonian of the system (the energy operator) and I is the reduced Planck constant 
(I = h/2p with h the Planck constant, allowing conversion from energy to frequency units). 
We will focus mainly on the Schr¨ odinger equation to describe the evolution of a quantum-mechanical system. The 
statement that the evolution of a closed quantum system is unitary is however more general. It means that the state 
of a system at a later time t is given by |?(t)) = U(t)|?(0)), where U(t) is a unitary operator. An operator is unitary 
if its adjoint U
† 
(obtained by taking the transpose and the complex conjugate of the operator, U
† 
= (U 
*
)
T 
) is equal 
U
-1
to its inverse: U
† 
= or UU
† 
= 1 1. 
Note that the expression |?(t)) = U(t)|?(0)) is an integral equation relating the state at time zero with the state at 
time t. For example, classically we could write that x(t) = x(0)+ vt (where v is the speed, for constant speed). We 
can as well write a di?erential equation that provides the same information: the Schr¨ odinger equation. Classically 
for example, (in the example above) the equivalent di?erential equation would be 
dx 
= v (more generally we would 
dt 
have Newton’s equation linking the acceleration to the force). In QM we have a di?erential equation that control the 
evolution of closed systems. This is the Schr¨ odinger equation: 
??(x, t)
iI = H?(x, t)
?t 
where H is the system’s Hamiltonian. The solution to this partial di?erential equation gives the wavefunction ?(x, t) 
at any later time, when ?(x, 0) is known. 
6.1.1 Solutions to the Schr o ¨dinger equation 
2 
p ˆ
We ?rst try to ?nd a solution in the case where the Hamiltonian H = +V (x, t) is such that the potential V (x, t)
2m 
is time independent (we can then write V (x)). In this case we can use separation of variables to look for solutions. 
That is, we look for solutions that are a product of a function of position only and a function of time only: 
?(x, t) = ?(x)f(t) 
83
Then, when we take the partial derivatives we have that 
??(x, t) df(t) ??(x, t) d?(x) ?
2
?(x, t) d
2
?(x) 
= ?(x), = f(t) and = f(t)
?t dt ?x dx ?x
2 
dx
2 
The Schr¨ odinger equation simpli?es to 
df(t) I
2 
d
2
?(x)
iI ?(x) = - f(t)+V (x)?(x)f(t)
dt 2m x
2 
Dividing by ?(x, t) we have: 
df(t) 1 I
2 
d
2
?(x) 1 
iI = - + V (x)
dt f(t) 2m x
2 
?(x) 
Now the LHS is a function of time only, while the RHS is a function of position only. For the equation to hold, both 
sides have then to be equal to a constant (separation constant): 
df(t) 1 I
2 
d
2
?(x) 1 
iI = E, - + V (x) = E 
dt f(t) 2m x
2 
?(x) 
The two equations we ?nd are a simple equation in the time variable: 
-i 
Et df(t) i 
i
= - Ef(t), ? f(t) = f(0)e 
dt I 
and 
I
2 
d
2
?(x) 1 
- + V (x) = E 
2m x
2 
?(x) 
that we have already seen as the time-independent Schr¨ odinger equation. We have extensively studied the solutions 
of the this last equation, as they are the eigenfunctions of the energy-eigenvalue problem, giving the stationary (equi­
librium) states of quantum systems. Note that for these stationary solutions ?(x) we can still ?nd the corresponding 
total wavefunction, given as stated above by ?(x, t) = ?(x)f(t), which does describe also the time evolution of the 
system: 
?(x, t) = ?(x)e 
-i 
Et
i 
Does this mean that the states that up to now we called stationary are instead evolving in time? 
The answerisyes,but witha caveat. Althoughthestatesthemselves evolve asstatedabove, any measurablequantity 
(such as the probability density |?(x, t)|
2 
or the expectation values of observable, (A) = 
J 
?(x, t)
*
A[?(x, t)]) are still 
time-independent. (Check it!) 
Thus we were correct in calling these states stationary and neglecting in practice their time-evolution when studying 
the properties of systems they describe. 
Notice that the wavefunction built from one energy eigenfunction, ?(x, t) = ?(x)f(t), is only a particular solution 
of the Schr¨ odinger equation, but many other are possible. These will be complicated functions of space and time, 
whose shape will depend on the particular form of the potential V (x). How can we describe these general solutions? 
We know that in general we can write a basis given by the eigenfunction of the Hamiltonian. These are the functions 
{?(x)} (as de?ned above by the time-independent Schr¨ odinger equation). The eigenstate of the Hamiltonian do not 
evolve. However we can write any wavefunction as 
L
?(x, t) = c
k
(t)?
k
(x) 
k 
This just corresponds to express the wavefunction in the basis given by the energy eigenfunctions. As usual, the 
coe?cients c
k
(t) can be obtained at any instant in time by taking the inner product: (?
k
|?(x, t)). 
What is the evolution of such a function? Substituting in the Schr¨ odinger equation we have 
?(
L
k 
c
k
(t)?
k
(x)) 
L
iI = c
k
(t)H?
k
(x)
?t 
k 
that becomes 
L 
?(c
k
(t)) 
L 
iI ?
k
(x) = c
k
(t)E
k
?
k
(x)
?t 
k k 
84
For each ?
k 
we then have the equation in the coe?cients only 
dc
k 
-i 
E
k
t 
i
iI = E
k
c
k
(t) ? c
k
(t) = c
k
(0)e 
dt 
A general solution of the Schr¨ odinger equation is then 
E
k
t
L 
-i 
i
?(x, t) = c
k
(0)e ?
k
(x) 
k 
Obs. We can de?ne the eigen-frequencies I?
k 
= E
k 
from the eigen-energies. Thus we see that the wavefunction is a 
superposition of waves ?
k 
propagating in time each with a di?erent frequency ?
k
. 
The behavior of quantum systems –even particles– thus often is similar to the propagation of waves. One example 
is the di?raction pattern for electrons (and even heavier objects) when scattering from a slit. We saw an example in 
the electron di?raction video at the beginning of the class. 
Obs. What is the probability of measuring a certain energy E
k 
at a time t? It is given by the coe?cient of the ?
k 
-i 
i
eigenfunction, |c
k
(t)|
2 
= |c
k
(0)e
E
k
t 
|
2 
= |c
k
(0)|
2
. This means that the probability for the given energy is constant, 
does not change in time. Energy is then a so-called constant of the motion. This is true only for the energy eigenvalues,
 
not for other observables‘.
 
Example: Considerinsteadtheprobabilityof?ndingthesystemata certainposition, p(x) = |?(x, t)|
2
. This of course
 
changes in time. For example, let ?(x, 0) = c
1
(0)?
1
(x)+c
2
(0)?
2
(x), with |c
1
(0)|
2 
+ |c
2
(0)|
2 
= |c
1
|
2 
+ |c
2
|
2 
= 1 (and
 
?
1,2 
normalized energy eigenfunctions. Then at a later time we have ?(x, 0) = c
1
(0)e
-i?1t
?
1
(x)+c
2
(0)e
-i?2t
?
2
(x).
 
What is p(x, t)?

 
-i?2 t
?
2
(x)
 
2 
 
c
1
(0)e 
-i?1t
?
1
(x)+c
2
(0)e 
 
* -i(?2-?1)t * i(?2-?1)t 
= |c
1
(0)|
2
|?
1
(x)|
2 
+ |c
2
(0)|
2
|?
2
(x)|
2 
+ c
1
c
2
?
1
* 
?
2
e + c
1
c
2
?
1
? 
* 
2
e 
[ ]
* -i(?2 -?1)t
= |c
1
|
2 
+ |c
2
|
2 
+ 2Rec
1
c
2
? 
* 
1
?
2
e 
The last term describes a wave interference between di?erent components of the initial wavefunction.
 
Obs.: The expressions found above for the time-dependent wavefunction are only valid if the potential is itself
 
time-independent. If this is not the case, the solutions are even more di?cult to obtain.
 
6.1.2 Unitary Evolution 
Wesawtwoequivalentformulationofthequantum mechanicalevolution,theSchr¨ odingerequationandthe Heisenberg 
equation. We now present a third possible formulation: following the 4
th 
postulate we express the evolution of a state 
in terms of a unitary operator, called the propagator: 
ˆ
?(x, t) = U(t)?(x, 0) 
with U
ˆ
†
U
ˆ
= 1 1. (Notice that a priori the unitary operator U
ˆ
could also be a function of space). We can show that 
this is equivalent to the Schr¨ odinger equation, by verifying that ?(x, t) above is a solution: 
? 
ˆ
? 
ˆ
U?(x, 0) U 
H 
ˆ
iI = U?(x, 0) ? iI = HU
ˆ
?t ?t 
where in the second step we used the fact that since the equation holds for any wavefunction ? it must hold for the 
operator themselves. If the Hamiltonian is time independent, the second equation can be solved easily, obtaining: 
?U
ˆ
-iHt/n
iI = HU
ˆ
? U
ˆ
(t) = e 
?t 
iHt/n -iHt/n
where we set U
ˆ
(t = 0) = 1 1. Notice that as desired U
ˆ
is unitary, U
ˆ
†
U
ˆ
= e e = 1 1. 
6.2 Evolution of wave-packets 
In Section 6.1.1 we looked at the evolution of a general wavefunction under a time-independent Hamiltonian. The 
solution to the Schr¨ odinger equation was given in terms of a linear superposition of energy eigenfunctions, each 
acquiring a time-dependent phase factor. The solution was then the superposition of waves each with a di?erent 
frequency. 
85
Now we want to study the case where the eigenfunctions form form a continuous basis, {?
k
} ? {?(k)}. More 
precisely, we want to describe how a free particle evolves in time. We already found the eigenfunctions of the free 
particle Hamiltonian(H = p ˆ
2
/2m):theyweregivenbythe momentum eigenfunctions e
ikx 
anddescribe moreproperly 
a traveling wave. A particle localized in space instead can be described by wavepacket ?(x, 0) initially well localized 
in x-space (for example, a Gaussian wavepacket). 
How does this wave-function evolve in time? First, following Section 2.2.1, we express the wavefunction in terms of 
momentum (and energy) eigenfunctions: 
1 
J
8 
?
¯
(k)e
ikx 
dk, ?(x, 0) = v 
2p 
-8 
¯
We saw that this is equivalent to the Fourier transform of ?
¯
(k), then ?(x, 0) and ?(k) are a Fourier pair (can be 
obtained from each other via a Fourier transform). 
¯
Thus the function ?(k)isobtainedbyFourier transformingthewave-function at t = 0. Notice again that the function 
¯
?(k) is the continuous-variable equivalent of the coe?cients c
k
(0). 
The second step is to evolve in time the superposition. From the previous section we know that each energy eigen­ function evolves by acquiring a phase e
-i?(k)t
, where ?(k) = E
k
/I is the energy eigenvalue. Then the time evolution 
of the wavefunction is 
J 
8 
¯
?(x, t) = ?(k)e
i?(k)
dk, 
-8 
where 
?(k) = kx - ?(k)t. 
nk
2 
For the free particle we have ?
k 
= . If the particle encounters instead a potential (such as in the potential barrier 
2m 
or potential well problems we already saw) ?
k 
could have a more complex form. We will thus consider this more 
general case. 
¯
Now, if ?(k) is strongly peaked around k = k
0
, it is a reasonable approximation to Taylor expand ?(k) about k
0
. 
(k-k
0
) 
2 
We can then approximate ?
¯
(k) by ?
¯
(k)˜ e 
-
4(?k)
2 
and keeping terms up to second-order in k - k
0
, we obtain 
J 
8 
(k-k
0
) 
2 
? { }? 
? 
'' 
?(x, t)? e 
-
4(?k)
2 
exp -ikx +i ?
0 
+ ? 
' 
0 
(k - k
0
)+ 
1 
0 
(k - k
0
)
2 
,
2
-8 
where 
?
0 
= ?(k
0
) = k
0 
x - ?
0 
t, 
d?(k0)
? 
' 
= = x - v
g 
t,
0 dk 
? 
'' 
d
2
?(k0)
= = -a t, 
0 dk
2 
{
1 
0 
(k - k
0
)
2
}
? 
'' 
-ikx +i k
0 
x - ?
0 
t +(x - v
g 
t)(k - k
0
)+ 
2 
with 
d?(k
0
) d
2
?(k
0
)
?
0 
= ?(k
0
), v
g 
= , a = . 
dk dk
2 
As usual, the variance of the initial wavefunction and of its Fourier transform are relates: ?k = 1/(2?x), where ?x 
is the initial width of the wave-packet and ?k the spread in the momentum. Changing the variable of integration to 
y = (k - k
0
)/(2?k), we get 
J 
8 
2
i(k0 x-?0 t) iß1 y-(1+iß2)y
?(x, t)? e e dy, 
-8 
where 
ß
1 
= 2?k (x - x
0 
- v
g 
t), 
ß
2 
= 2a (?k)
2 
t, 
The above expression can be rearranged to give 
J
8 
i(k
0 x-?0 t)-(1+iß2)ß
2
/4 -(1+iß2)(y-y0)
2 
?(x, t)? e e dy, 
-8 
where y
0 
= iß/2 and ß = ß
1
/(1+iß
2
).
 
Again changing the variable of integration to z = (1+i ß
2
)
1/2 
(y - y
0
) , we get
 
J 
8 
i(k
0 x-?0 t)-(1+iß2 )ß 
2
/4 -z
?(x, t)? (1+iß
2
)
-1/2 
e e 
2 
dz. 
-8 
86
Page 5


6. Time Evolution in Quantum Mechanics 
6.1 Time-dependent Schr¨ odinger equation 
6.1.1 Solutions to the Schr¨ odinger equation 
6.1.2 Unitary Evolution 
6.2 Evolution of wave-packets 
6.3 Evolution of operators and expectation values 
6.3.1 Heisenberg Equation 
6.3.2 Ehrenfest’s theorem 
6.4 Fermi’s Golden Rule 
Until now we used quantum mechanics to predict properties of atoms and nuclei. Since we were interested mostly 
in the equilibrium states of nuclei and in their energies, we only needed to look at a time-independent description 
of quantum-mechanical systems. To describe dynamical processes, such as radiation decays, scattering and nuclear 
reactions, we need to study how quantum mechanical systems evolve in time. 
6.1 Time-dependent Schr o ¨dinger equation 
When we ?rst introduced quantum mechanics, we saw that the fourth postulate of QM states that:
 
The evolution of a closed system is unitary (reversible). The evolution is given by the time-dependent Schr ¨
 odinger 
equation 
?|?)
iI = H|?)
?t 
where H is the Hamiltonian of the system (the energy operator) and I is the reduced Planck constant 
(I = h/2p with h the Planck constant, allowing conversion from energy to frequency units). 
We will focus mainly on the Schr¨ odinger equation to describe the evolution of a quantum-mechanical system. The 
statement that the evolution of a closed quantum system is unitary is however more general. It means that the state 
of a system at a later time t is given by |?(t)) = U(t)|?(0)), where U(t) is a unitary operator. An operator is unitary 
if its adjoint U
† 
(obtained by taking the transpose and the complex conjugate of the operator, U
† 
= (U 
*
)
T 
) is equal 
U
-1
to its inverse: U
† 
= or UU
† 
= 1 1. 
Note that the expression |?(t)) = U(t)|?(0)) is an integral equation relating the state at time zero with the state at 
time t. For example, classically we could write that x(t) = x(0)+ vt (where v is the speed, for constant speed). We 
can as well write a di?erential equation that provides the same information: the Schr¨ odinger equation. Classically 
for example, (in the example above) the equivalent di?erential equation would be 
dx 
= v (more generally we would 
dt 
have Newton’s equation linking the acceleration to the force). In QM we have a di?erential equation that control the 
evolution of closed systems. This is the Schr¨ odinger equation: 
??(x, t)
iI = H?(x, t)
?t 
where H is the system’s Hamiltonian. The solution to this partial di?erential equation gives the wavefunction ?(x, t) 
at any later time, when ?(x, 0) is known. 
6.1.1 Solutions to the Schr o ¨dinger equation 
2 
p ˆ
We ?rst try to ?nd a solution in the case where the Hamiltonian H = +V (x, t) is such that the potential V (x, t)
2m 
is time independent (we can then write V (x)). In this case we can use separation of variables to look for solutions. 
That is, we look for solutions that are a product of a function of position only and a function of time only: 
?(x, t) = ?(x)f(t) 
83
Then, when we take the partial derivatives we have that 
??(x, t) df(t) ??(x, t) d?(x) ?
2
?(x, t) d
2
?(x) 
= ?(x), = f(t) and = f(t)
?t dt ?x dx ?x
2 
dx
2 
The Schr¨ odinger equation simpli?es to 
df(t) I
2 
d
2
?(x)
iI ?(x) = - f(t)+V (x)?(x)f(t)
dt 2m x
2 
Dividing by ?(x, t) we have: 
df(t) 1 I
2 
d
2
?(x) 1 
iI = - + V (x)
dt f(t) 2m x
2 
?(x) 
Now the LHS is a function of time only, while the RHS is a function of position only. For the equation to hold, both 
sides have then to be equal to a constant (separation constant): 
df(t) 1 I
2 
d
2
?(x) 1 
iI = E, - + V (x) = E 
dt f(t) 2m x
2 
?(x) 
The two equations we ?nd are a simple equation in the time variable: 
-i 
Et df(t) i 
i
= - Ef(t), ? f(t) = f(0)e 
dt I 
and 
I
2 
d
2
?(x) 1 
- + V (x) = E 
2m x
2 
?(x) 
that we have already seen as the time-independent Schr¨ odinger equation. We have extensively studied the solutions 
of the this last equation, as they are the eigenfunctions of the energy-eigenvalue problem, giving the stationary (equi­
librium) states of quantum systems. Note that for these stationary solutions ?(x) we can still ?nd the corresponding 
total wavefunction, given as stated above by ?(x, t) = ?(x)f(t), which does describe also the time evolution of the 
system: 
?(x, t) = ?(x)e 
-i 
Et
i 
Does this mean that the states that up to now we called stationary are instead evolving in time? 
The answerisyes,but witha caveat. Althoughthestatesthemselves evolve asstatedabove, any measurablequantity 
(such as the probability density |?(x, t)|
2 
or the expectation values of observable, (A) = 
J 
?(x, t)
*
A[?(x, t)]) are still 
time-independent. (Check it!) 
Thus we were correct in calling these states stationary and neglecting in practice their time-evolution when studying 
the properties of systems they describe. 
Notice that the wavefunction built from one energy eigenfunction, ?(x, t) = ?(x)f(t), is only a particular solution 
of the Schr¨ odinger equation, but many other are possible. These will be complicated functions of space and time, 
whose shape will depend on the particular form of the potential V (x). How can we describe these general solutions? 
We know that in general we can write a basis given by the eigenfunction of the Hamiltonian. These are the functions 
{?(x)} (as de?ned above by the time-independent Schr¨ odinger equation). The eigenstate of the Hamiltonian do not 
evolve. However we can write any wavefunction as 
L
?(x, t) = c
k
(t)?
k
(x) 
k 
This just corresponds to express the wavefunction in the basis given by the energy eigenfunctions. As usual, the 
coe?cients c
k
(t) can be obtained at any instant in time by taking the inner product: (?
k
|?(x, t)). 
What is the evolution of such a function? Substituting in the Schr¨ odinger equation we have 
?(
L
k 
c
k
(t)?
k
(x)) 
L
iI = c
k
(t)H?
k
(x)
?t 
k 
that becomes 
L 
?(c
k
(t)) 
L 
iI ?
k
(x) = c
k
(t)E
k
?
k
(x)
?t 
k k 
84
For each ?
k 
we then have the equation in the coe?cients only 
dc
k 
-i 
E
k
t 
i
iI = E
k
c
k
(t) ? c
k
(t) = c
k
(0)e 
dt 
A general solution of the Schr¨ odinger equation is then 
E
k
t
L 
-i 
i
?(x, t) = c
k
(0)e ?
k
(x) 
k 
Obs. We can de?ne the eigen-frequencies I?
k 
= E
k 
from the eigen-energies. Thus we see that the wavefunction is a 
superposition of waves ?
k 
propagating in time each with a di?erent frequency ?
k
. 
The behavior of quantum systems –even particles– thus often is similar to the propagation of waves. One example 
is the di?raction pattern for electrons (and even heavier objects) when scattering from a slit. We saw an example in 
the electron di?raction video at the beginning of the class. 
Obs. What is the probability of measuring a certain energy E
k 
at a time t? It is given by the coe?cient of the ?
k 
-i 
i
eigenfunction, |c
k
(t)|
2 
= |c
k
(0)e
E
k
t 
|
2 
= |c
k
(0)|
2
. This means that the probability for the given energy is constant, 
does not change in time. Energy is then a so-called constant of the motion. This is true only for the energy eigenvalues,
 
not for other observables‘.
 
Example: Considerinsteadtheprobabilityof?ndingthesystemata certainposition, p(x) = |?(x, t)|
2
. This of course
 
changes in time. For example, let ?(x, 0) = c
1
(0)?
1
(x)+c
2
(0)?
2
(x), with |c
1
(0)|
2 
+ |c
2
(0)|
2 
= |c
1
|
2 
+ |c
2
|
2 
= 1 (and
 
?
1,2 
normalized energy eigenfunctions. Then at a later time we have ?(x, 0) = c
1
(0)e
-i?1t
?
1
(x)+c
2
(0)e
-i?2t
?
2
(x).
 
What is p(x, t)?

 
-i?2 t
?
2
(x)
 
2 
 
c
1
(0)e 
-i?1t
?
1
(x)+c
2
(0)e 
 
* -i(?2-?1)t * i(?2-?1)t 
= |c
1
(0)|
2
|?
1
(x)|
2 
+ |c
2
(0)|
2
|?
2
(x)|
2 
+ c
1
c
2
?
1
* 
?
2
e + c
1
c
2
?
1
? 
* 
2
e 
[ ]
* -i(?2 -?1)t
= |c
1
|
2 
+ |c
2
|
2 
+ 2Rec
1
c
2
? 
* 
1
?
2
e 
The last term describes a wave interference between di?erent components of the initial wavefunction.
 
Obs.: The expressions found above for the time-dependent wavefunction are only valid if the potential is itself
 
time-independent. If this is not the case, the solutions are even more di?cult to obtain.
 
6.1.2 Unitary Evolution 
Wesawtwoequivalentformulationofthequantum mechanicalevolution,theSchr¨ odingerequationandthe Heisenberg 
equation. We now present a third possible formulation: following the 4
th 
postulate we express the evolution of a state 
in terms of a unitary operator, called the propagator: 
ˆ
?(x, t) = U(t)?(x, 0) 
with U
ˆ
†
U
ˆ
= 1 1. (Notice that a priori the unitary operator U
ˆ
could also be a function of space). We can show that 
this is equivalent to the Schr¨ odinger equation, by verifying that ?(x, t) above is a solution: 
? 
ˆ
? 
ˆ
U?(x, 0) U 
H 
ˆ
iI = U?(x, 0) ? iI = HU
ˆ
?t ?t 
where in the second step we used the fact that since the equation holds for any wavefunction ? it must hold for the 
operator themselves. If the Hamiltonian is time independent, the second equation can be solved easily, obtaining: 
?U
ˆ
-iHt/n
iI = HU
ˆ
? U
ˆ
(t) = e 
?t 
iHt/n -iHt/n
where we set U
ˆ
(t = 0) = 1 1. Notice that as desired U
ˆ
is unitary, U
ˆ
†
U
ˆ
= e e = 1 1. 
6.2 Evolution of wave-packets 
In Section 6.1.1 we looked at the evolution of a general wavefunction under a time-independent Hamiltonian. The 
solution to the Schr¨ odinger equation was given in terms of a linear superposition of energy eigenfunctions, each 
acquiring a time-dependent phase factor. The solution was then the superposition of waves each with a di?erent 
frequency. 
85
Now we want to study the case where the eigenfunctions form form a continuous basis, {?
k
} ? {?(k)}. More 
precisely, we want to describe how a free particle evolves in time. We already found the eigenfunctions of the free 
particle Hamiltonian(H = p ˆ
2
/2m):theyweregivenbythe momentum eigenfunctions e
ikx 
anddescribe moreproperly 
a traveling wave. A particle localized in space instead can be described by wavepacket ?(x, 0) initially well localized 
in x-space (for example, a Gaussian wavepacket). 
How does this wave-function evolve in time? First, following Section 2.2.1, we express the wavefunction in terms of 
momentum (and energy) eigenfunctions: 
1 
J
8 
?
¯
(k)e
ikx 
dk, ?(x, 0) = v 
2p 
-8 
¯
We saw that this is equivalent to the Fourier transform of ?
¯
(k), then ?(x, 0) and ?(k) are a Fourier pair (can be 
obtained from each other via a Fourier transform). 
¯
Thus the function ?(k)isobtainedbyFourier transformingthewave-function at t = 0. Notice again that the function 
¯
?(k) is the continuous-variable equivalent of the coe?cients c
k
(0). 
The second step is to evolve in time the superposition. From the previous section we know that each energy eigen­ function evolves by acquiring a phase e
-i?(k)t
, where ?(k) = E
k
/I is the energy eigenvalue. Then the time evolution 
of the wavefunction is 
J 
8 
¯
?(x, t) = ?(k)e
i?(k)
dk, 
-8 
where 
?(k) = kx - ?(k)t. 
nk
2 
For the free particle we have ?
k 
= . If the particle encounters instead a potential (such as in the potential barrier 
2m 
or potential well problems we already saw) ?
k 
could have a more complex form. We will thus consider this more 
general case. 
¯
Now, if ?(k) is strongly peaked around k = k
0
, it is a reasonable approximation to Taylor expand ?(k) about k
0
. 
(k-k
0
) 
2 
We can then approximate ?
¯
(k) by ?
¯
(k)˜ e 
-
4(?k)
2 
and keeping terms up to second-order in k - k
0
, we obtain 
J 
8 
(k-k
0
) 
2 
? { }? 
? 
'' 
?(x, t)? e 
-
4(?k)
2 
exp -ikx +i ?
0 
+ ? 
' 
0 
(k - k
0
)+ 
1 
0 
(k - k
0
)
2 
,
2
-8 
where 
?
0 
= ?(k
0
) = k
0 
x - ?
0 
t, 
d?(k0)
? 
' 
= = x - v
g 
t,
0 dk 
? 
'' 
d
2
?(k0)
= = -a t, 
0 dk
2 
{
1 
0 
(k - k
0
)
2
}
? 
'' 
-ikx +i k
0 
x - ?
0 
t +(x - v
g 
t)(k - k
0
)+ 
2 
with 
d?(k
0
) d
2
?(k
0
)
?
0 
= ?(k
0
), v
g 
= , a = . 
dk dk
2 
As usual, the variance of the initial wavefunction and of its Fourier transform are relates: ?k = 1/(2?x), where ?x 
is the initial width of the wave-packet and ?k the spread in the momentum. Changing the variable of integration to 
y = (k - k
0
)/(2?k), we get 
J 
8 
2
i(k0 x-?0 t) iß1 y-(1+iß2)y
?(x, t)? e e dy, 
-8 
where 
ß
1 
= 2?k (x - x
0 
- v
g 
t), 
ß
2 
= 2a (?k)
2 
t, 
The above expression can be rearranged to give 
J
8 
i(k
0 x-?0 t)-(1+iß2)ß
2
/4 -(1+iß2)(y-y0)
2 
?(x, t)? e e dy, 
-8 
where y
0 
= iß/2 and ß = ß
1
/(1+iß
2
).
 
Again changing the variable of integration to z = (1+i ß
2
)
1/2 
(y - y
0
) , we get
 
J 
8 
i(k
0 x-?0 t)-(1+iß2 )ß 
2
/4 -z
?(x, t)? (1+iß
2
)
-1/2 
e e 
2 
dz. 
-8 
86
? ?
The integral now just reduces to a number. Hence, we obtain 
(x-x
0
-v
g 
t)
2
[1-i2 a?k
2
t] 
e
i(k0 x-?0 t)
e 
-
4 s(t)
2 
?(x, t)? , ?
1+i2a (?k)
2 
t 
where 
a
2 
t
2 
s
2
(t) = (?x)
2 
+ . 
4(?x)
2
Note that even if we made an approximation earlier by Taylor expanding the phase factor ?(k) about k = k
0
, the 
above wave-function is still identical to our original wave-function at t = 0. 
The probability density of our particle as a function of times is written 
(x - x
0 
- v
g 
t)
2 
|?(x, t)| 
2 
? s
-1
(t)exp - . 
2s 
2
(t) 
Hence, the probability distribution is a Gaussian, of characteristic width s(t) (increasing in time), which peaks at 
x = x
0 
+ v
g 
t. Now, the most likely position of our particle obviously coincides with the peak of the distribution 
function. Thus, the particle’s most likely position is given by 
x = x
0 
+ v
g 
t. 
It can be seen that the particle e?ectively moves at the uniform velocity 
d? 
v
g 
= ,
dk 
which isknown as the group-velocity.In otherwords,aplane-wavetravelsatthephase-velocity, v
p 
= ?/k, whereas 
a wave-packet travels at the group-velocity, v
g 
= d?/dt v
g 
= d?/dt. From the dispersion relation for particle waves 
the group velocity is 
d(I?) dE p 
v
g 
= = = . 
d(Ik) dp m 
which is identical to the classical particle velocity. Hence, the dispersion relation turns out to be consistent with 
classical physics, after all, as soon as we realize that particles must be identi?ed with wave-packets rather than 
plane-waves. 
Note that the width of our wave-packet grows as time progresses: the characteristic time for a wave-packet of original 
width ?x ?x to double in spatial extent is 
m (?x)
2 
t
2 
~ . 
I 
So, if an electron is originally localized in a region of atomic scale (i.e., ?x ~ 10
-10 
m ) then the doubling time is
 
only about 10
-16
s. Clearly, particle wave-packets (for freely moving particles) spread very rapidly.
 
The rate of spreading of a wave-packet is ultimately governed by the second derivative of ?(k) with respect to k,
 
?
2
? 
?k
2 
. This is why the relationship between ? and k is generally known as a dispersion relation, because it governs
 
how wave-packets disperse as time progresses.
 
If we consider light-waves, then ? is a linear function of k and the second derivative of ? with respect to k is zero.
 
This implies that there is no dispersion of wave-packets, wave-packets propagate without changing shape. This is
 
of course true for any other wave for which ?(k) ? k. Another property of linear dispersion relations is that the
 
phase-velocity, v
p 
= ?/k, and the group-velocity, v
g 
= d?/dk are identical. Thus a light pulse propagates at the
 
same speed of a plane light-wave; both propagate through a vacuum at the characteristic speed c = 3× 10
8 
m/s .
 
Of course, the dispersion relation for particle waves is not linear in k (for example for free particles is quadratic).
 
Hence, particle plane-waves and particle wave-packets propagate at di?erent velocities, and particle wave-packets
 
also gradually disperse as time progresses.
 
6.3 Evolution of operators and expectation values 
The Schr¨ odinger equation describes how the state of a system evolves. Since via experiments we have access to 
observables and their outcomes, it is interesting to ?nd a di?erential equation that directly gives the evolution of 
expectation values. 


87
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