The Schrödinger Equation
• Throughout this modules we are going to deal with a non-relativistic
quantum mechanics.
• In order to achieve that goal, we need to set up a wave equation required to
describe a wave-particle property of quantum particles.
• The equation contemplated is the Schrӧdinger wave equation.
• The role played by Schrӧdinger equation in quantum mechanics is the same
as the role played by Newton’s Laws equations in classical mechanics.
• Both equations describe the motion.
• Newtons’ 2nd Law is a second order differential equation which describes
how a classical particle moves in space:
𝐹𝐹⃗ = 𝑚𝑚𝑎𝑎⃗ = 𝑚𝑚𝑟𝑟.
⃗̈
• Schrӧdinger equation is a second order partial differential equation which
describes how the wavefunction corresponding to a particle propagates in
space.
• It must be mentioned that both the equations were postulated and then
experiments were performed to test their validity.
Sinusoidal Waves
• The well-designed wave is a sinusoidal wave travelling with wavelength 𝜆𝜆 and period 𝜏𝜏.
• In the same way this can be represented by the wave number 𝑘𝑘 = 2𝜋𝜋⁄𝜆𝜆 and angular
frequency 𝜔𝜔 = 2𝜋𝜋⁄𝜏𝜏.
• Such a wave is represented by a mathematical function
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 cos 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 .
• 𝐴𝐴 is a constant for the amplitude.
2𝜋𝜋
• At point 𝑥𝑥, the function oscillates with amplitude 𝐴𝐴 and period .
𝜔𝜔
2𝜋𝜋
• At time 𝑡𝑡, the function undulates with amplitude 𝐴𝐴 and wavelength .
𝑘𝑘
• On top of that these undulations the wave function exhibit a Mexican wave, in the direction
of the increasing 𝑥𝑥 with velocity 𝜔𝜔/𝑘𝑘.
• For the case in point, the maximum of 𝜓𝜓(𝑥𝑥, 𝑡𝑡) corresponds to
𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 = 0.
• This occurs at position 𝑥𝑥 = 𝜔𝜔𝜔𝜔/𝑘𝑘.
𝜆𝜆
• The minimum of 𝜓𝜓(𝑥𝑥, 𝑡𝑡) corresponds to 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 = 𝜋𝜋 and it occurs at position 𝑥𝑥 = +
2
• In both cases the position moves with velocity 𝜔𝜔/𝑘𝑘.
𝜔𝜔𝜔𝜔
.
𝑘𝑘
• The function sin 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 , like cos 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 , also represents a sinusoidal
travelling wave with wave number 𝑘𝑘 and angular frequency 𝜔𝜔.
• Since we know that sin 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 = cos 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 − 𝜋𝜋/2
• The undulations and oscillations of sin 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 are out of step in relation to
those of cos 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 .
• So the waves of sin 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 and cos 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 have a phase difference of
𝜋𝜋/2.
• Now the most general sinusoidal travelling wave having wave number 𝑘𝑘 and
angular frequency 𝜔𝜔 is the linear superposition of the two waves to give:
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴 cos(𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔) + 𝐵𝐵 sin(𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔),
• 𝐴𝐴 and 𝐵𝐵 are constants.
• Occasionally in classical physics and invariably in quantum physics, sinusoidal
travelling waves are represented by the complex exponential function
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴𝑒𝑒 𝑖𝑖(𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔) .
• 𝐴𝐴𝑒𝑒 𝑖𝑖(𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔) = 𝐴𝐴[cos 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 + 𝑖𝑖 sin(𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔)].
Linear superposition of sinusoidal waves
• Two sinusoidal waves travelling in opposite directions may be combined to form standing waves.
• For instance, the linear superposition
𝐴𝐴 cos 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 + 𝐴𝐴 cos(𝑘𝑘𝑘𝑘 + 𝜔𝜔𝜔𝜔)
• give rise to the wave
2𝐴𝐴 cos 𝑘𝑘𝑘𝑘 cos 𝜔𝜔𝜔𝜔.
• This wave oscillates with period 2𝜋𝜋/𝜔𝜔 and undulates with wavelength 2𝜋𝜋/𝑘𝑘.
• But these oscillations and undulations do not propagate, they are just standing waves.
• Many sinusoidal waves may be combined to form a wave packet.
• For example, the mathematical form of a wave packet formed by linear superposition of sinusoidal
waves with constant amplitude 𝐴𝐴 and wave numbers in the range 𝑘𝑘 − Δ𝑘𝑘 to 𝑘𝑘 + Δ𝑘𝑘 is
𝑘𝑘+Δ𝑘𝑘
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = ∫𝑘𝑘−Δ𝑘𝑘 𝐴𝐴 cos 𝑘𝑘′𝑥𝑥 − 𝜔𝜔′𝑡𝑡 𝑑𝑑𝑑𝑑′.
• If 𝑘𝑘 is positive, the wave packet travels in the positive 𝑥𝑥 direction, and to the negative 𝑥𝑥 direction when 𝑘𝑘
in negative.
• The initial shape of the wave packet, i.e. a shape at 𝑡𝑡 = 0, may be obtained by evaluating the integral
𝑘𝑘+Δ𝑘𝑘
𝜓𝜓 𝑥𝑥, 0 = ∫𝑘𝑘−Δ𝑘𝑘 𝐴𝐴 cos 𝑘𝑘′𝑥𝑥𝑥𝑥𝑥𝑥′
• this gives 𝜓𝜓 𝑥𝑥, 0 = 𝒮𝒮 𝑥𝑥 cos 𝑘𝑘𝑘𝑘 where 𝒮𝒮 𝑥𝑥 = 2𝐴𝐴∆𝑘𝑘
sin ∆𝑘𝑘𝑘𝑘
.
(∆𝑘𝑘𝑘𝑘)
Dispersive and non-dispersive waves
• A good example of non-dispersive waves are electromagnetic waves travelling in a
vacuum.
• A non-dispersive wave has a dispersion relation of the form 𝜔𝜔 = 𝑐𝑐𝑐𝑐
• The velocity, 𝑐𝑐 = 𝜔𝜔/𝑘𝑘, is independent of the wave number 𝑘𝑘.
• A wave packet formed from such sinusoidal waves travels without changing shape
because each sinusoidal component has the same velocity.
• The non-dispersive waves are described by a partial differential equation called
classical wave equation.
• In one dimension along the 𝑥𝑥 direction, the wave equation has the form
• In three
𝜕𝜕2 𝜓𝜓
1 𝜕𝜕2 𝜓𝜓
= 2 2,
𝜕𝜕𝜕𝜕 2
𝑐𝑐 𝜕𝜕𝜕𝜕 2
1 𝜕𝜕 𝜓𝜓
2
dimensions 𝛻𝛻 𝜓𝜓 = 2 2 ,
𝑐𝑐 𝜕𝜕𝜕𝜕
2
where 𝛻𝛻 =
𝜕𝜕2
𝜕𝜕𝜕𝜕 2
𝜕𝜕2
+ 2
𝜕𝜕𝜕𝜕
𝜕𝜕2
+ 2.
𝜕𝜕𝜕𝜕
• The classical wave equation has infinite number of solutions which corresponds to
different wave forms.
• e.g. the sinusoidal waves can be
𝐴𝐴 cos(𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔), 𝐴𝐴 sin(𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔), or 𝐴𝐴𝑒𝑒 𝑖𝑖(𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔)
• Under normal circumstances, most of the waves encountered in classical and quantum
physics are dispersive waves.
• The partial differential equation describing dispersive waves is more complicated than the
one discussed for classical wave equation.
• The dispersion relation, 𝜔𝜔 = 𝑐𝑐𝑐𝑐 is more complicated so that the velocity, 𝜔𝜔/𝑘𝑘 of a
propagating wave depends on the wave number, 𝑘𝑘.
• This explains why the packet of the dispersive waves generally does change in shape as they
propagate in space.
• But, if the packet consists of a narrow range of wave numbers, it can have a well-defined
velocity of propagation.
• This velocity is called the group velocity and is given by
𝑑𝑑𝑑𝑑
𝜐𝜐𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 =
𝑑𝑑𝑑𝑑
• But the velocity of a simple sinusoidal wave is called the phase velocity
𝜔𝜔
𝜐𝜐𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = .
𝑘𝑘
• The group velocity describes the motion of a localized disturbance resulting from
constructive interference of two or more sinusoidal waves.
• e.g. a constructive interference resulting from two sinusoidal waves with wave numbers 𝑘𝑘1
and 𝑘𝑘2 and angular frequencies 𝜔𝜔1 and 𝜔𝜔2 , mathematically this is
𝑘𝑘1 𝑥𝑥 − 𝜔𝜔1 𝑡𝑡 = 𝑘𝑘2 𝑥𝑥 − 𝜔𝜔2 𝑡𝑡
• The point of constructive interference is obtained by rearranging the above equation to give
𝑥𝑥 =
𝜔𝜔1 −𝜔𝜔2
𝑘𝑘1 −𝑘𝑘2
𝑡𝑡 .
• So, the point of constructive interference is located at 𝑥𝑥 = 0, when 𝑡𝑡 = 0 and it travels with
a velocity given by 𝜔𝜔1 − 𝜔𝜔2 /𝑘𝑘1 − 𝑘𝑘2
• As an example, consider water waves of long wavelength obeying the dispersion relation
𝜔𝜔 = 𝑔𝑔𝑔𝑔 ,
• 𝑔𝑔 is the acceleration due to gravity.
• The phase velocity of the sinusoidal wave is
𝜐𝜐𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 =
𝜔𝜔
𝑘𝑘
=
𝑔𝑔
𝑘𝑘
.
• The velocity of the wave packet, with a narrow range of wave numbers near 𝑘𝑘 is
𝜐𝜐𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
=
1
2
𝑔𝑔
𝑘𝑘
.
• So for water waves the group velocity is one-half of the phase velocity.
• In other words, the sinusoidal waves forming a wave packet travel at twice the speed of the
region of maximum disturbance caused by the interference of these waves.
• The shape of the disturbance will change as it propagates, which means it tends to spread out.
Initial shapes demonstrating superposition of sinusoidal waves in the region k-Δk and k+Δk
Free particle wave equation
• we need to construct a possible wave equation for a freely moving non-relativistic
particle by considering the properties of the de Broglie’s waves.
• a particle with momentum 𝑝𝑝 has de Broglie wavelength given by 𝜆𝜆 = ℎ/𝑝𝑝.
• this means that a de Broglie wave with wave number 𝑘𝑘 = 2𝜋𝜋/𝜆𝜆 describes a particle
with momentum
ℎ
𝑝𝑝 = ℏ𝑘𝑘 , where
ℏ=
2𝜋𝜋
• to extend this argument, assume that a de Broglie wave packet with a range of wave
numbers from 𝑘𝑘 − Δ𝑘𝑘
to 𝑘𝑘 + Δ𝑘𝑘
describes a particle with an uncertain
momentum
Δ𝑝𝑝 ≈ ℏ∆𝑘𝑘
• assume also that the length of this wave packet is a measure of Δ𝑥𝑥, the uncertainty
in the position of the particle.
2𝜋𝜋
∆𝑥𝑥 ≈
∆𝑘𝑘
• using the equation Δ𝑝𝑝 ≈ ℏ∆𝑘𝑘 and the figure of linear superposition of waves as a
model
• we multiply these two uncertainties to obtain
∆𝑥𝑥∆𝑝𝑝 ≈ ℎ
• this means a de Broglie wave packet can account for the uncertainties in the position and
momentum of a quantum particle.
• this also means that a de Broglie wave is transformed by a measurement.
• if a precise measurement of the position is made, the new packet describing the particle
will be very short, a superposition of sinusoidal waves with a wide range of wavelengths.
• if a precise measurement of momentum is made, the new wave packet will be long with a
well defined wavelength.
• this shows that a wave packet is a delicate entity which is transformed by measurements.
• the notion that the wave packet represents a moving quantum particle is being imposed.
• the fact that the group velocity of the packet equals the velocity of a particle with mass 𝑚𝑚
and momentum 𝑝𝑝 = ℏ𝑘𝑘 requires that
•
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
=
ℏ𝑘𝑘
𝑚𝑚
• this equation can be integrated to give the following dispersion relation for
the de Broglie waves describing a free moving quantum particle of mass 𝑚𝑚:
𝜔𝜔 =
ℏ𝑘𝑘 2
.
2𝑚𝑚
• in order to obtain the above relation the constant of integration is set to zero.
• in any case this constant give rise to no observable consequences in the nonrelativistic quantum mechanics.
• the main goal is to obtain a wave equation with sinusoidal solutions obeying
the above dispersion relation.
• the simplest such a wave equation is called the Schrӧdinger equation.
• for a free particle moving in one dimension, it has the form
𝜕𝜕𝜕𝜕
𝑖𝑖𝑖
𝜕𝜕𝜕𝜕
=
ℏ2 𝜕𝜕2 𝜓𝜓
−
2𝑚𝑚 𝜕𝜕𝜕𝜕 2
• the complex exponential function 𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴𝑒𝑒 𝑖𝑖(𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔)
is the
ℏ𝑘𝑘 2
solution of this equation provided it obey the dispersion relation 𝜔𝜔 =
.
2𝑚𝑚
• substituting 𝜓𝜓 𝑥𝑥, 𝑡𝑡 on the left hand side of the Schrödinger eqn yields
𝜕𝜕𝜕𝜕
𝑖𝑖𝑖
𝜕𝜕𝜕𝜕
= 𝑖𝑖𝑖 −𝑖𝑖𝑖𝑖 𝐴𝐴𝑒𝑒 𝑖𝑖(𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔) = ℏ𝜔𝜔𝜔𝜔𝑒𝑒 𝑖𝑖(𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔) = ℏ𝜔𝜔𝜔𝜔(𝑥𝑥, 𝑡𝑡)
• the right hand side yields
ℏ2 𝜕𝜕2 𝜓𝜓
−
2𝑚𝑚 𝜕𝜕𝜕𝜕 2
ℏ2 𝜕𝜕2
𝑖𝑖 𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔
=−
𝐴𝐴𝑒𝑒
2
2 2𝑚𝑚 𝜕𝜕𝜕𝜕
ℏ
−
−𝑘𝑘 2 𝐴𝐴𝑒𝑒 𝑖𝑖 𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔
2𝑚𝑚
• the solution can be obtained provided
ℏ𝜔𝜔 =
ℏ2 𝑘𝑘 2
2𝑚𝑚
=
ℏ2 𝑘𝑘 2
=
𝜓𝜓(𝑥𝑥, 𝑡𝑡)
2𝑚𝑚
• the sinusoidal wave solution, 𝐴𝐴𝑒𝑒 𝑖𝑖(𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔) , describes a wave moving in the xdirection with wave number 𝑘𝑘 and angular frequency 𝜔𝜔.
• we assume that the wave represents a free 2particle moving in the x-direction
𝑝𝑝
= ℏ𝜔𝜔.
with momentum 𝑝𝑝 = ℏ𝑘𝑘 and energy 𝐸𝐸 =
2𝑚𝑚
• of course, there are many other solutions of the Schrӧdinger equation
representing other states of motion of the particle.
• recall that classically waves are often represented by complex functions, as a
matter of mathematical convenience.
• in the practical sense, classical waves normally are real functions of space and time.
• alternatively, Schrӧdinger wave functions are not real functions of space and time.
• they are complex functions which describe the concealed wave-like property of a
quantum particle.
• a superposition of solutions is also a solution. e.g.
•
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴1 𝑒𝑒 𝑖𝑖(𝑘𝑘1 𝑥𝑥−𝜔𝜔1 𝑡𝑡) + 𝐴𝐴2 𝑒𝑒 𝑖𝑖(𝑘𝑘2 𝑥𝑥−𝜔𝜔2 𝑡𝑡)
•
•
•
•
ℏ2 𝑘𝑘12
2𝑚𝑚
ℏ2 𝑘𝑘22
,
2𝑚𝑚
with
ℏ𝜔𝜔1 =
and
ℏ𝜔𝜔2 =
here 𝐴𝐴1 and 𝐴𝐴2 are arbitrary constants,
this can easily be confirmed by direct substitution.
The most general solution is the superposition of sinusoidal waves with all possible
wave numbers and angular frequencies:
+∞
′ 𝑖𝑖 𝑘𝑘 ′ 𝑥𝑥−𝜔𝜔′ 𝑡𝑡
∫−∞ 𝐴𝐴(𝑘𝑘 )𝑒𝑒
ℏ2 𝑘𝑘′2
.
2𝑚𝑚
•
𝜓𝜓 𝑥𝑥, 𝑡𝑡 =
𝑑𝑑𝑑𝑑′ with ℏ𝜔𝜔′ =
• Here, 𝐴𝐴(𝑘𝑘 ′ ) is an arbitrary complex function of 𝑘𝑘′, and the integral is over all
possible wave numbers 𝑘𝑘′.
SPHA031 Worked QM Examples 2021
A free electron has the wavefunction
𝜓𝜓 𝑥𝑥, 𝑡𝑡 = sin(𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔)
and wave number 𝑘𝑘 = 50nm−1
(a) Determine the electron’s de Broglie wavelength.
2𝜋𝜋
𝑘𝑘 =
𝜆𝜆
2𝜋𝜋 2𝜋𝜋
=
nm = 0.126 nm = 126 pm
𝜆𝜆 =
𝑘𝑘
50
(b) Determine the electron’s momentum.
ℎ
𝑝𝑝 =
𝜆𝜆
ℎ 2𝜋𝜋
=
𝜆𝜆 2𝜋𝜋
= ℏ𝑘𝑘
=
=
𝑐𝑐
ℏ𝑘𝑘
𝑐𝑐
197.32eVm (50nm−1 )
𝑐𝑐
= 9866 eV/c
= 9.87 keV/c
(c) Determine the electron’s kinetic energy.
ℏ2 𝑘𝑘 2
𝐸𝐸 =
2𝑚𝑚
=
=
=
𝑝𝑝2
2𝑚𝑚
9.87keV/c 2
2 511keV/𝑐𝑐 2
97.417
keV
3
1.022×10
9.53 × 10−2 keV
=
= 95.3eV
(d) Determine the electron’s speed.
𝑑𝑑𝑑𝑑 𝑝𝑝
=
𝜐𝜐 = 𝜐𝜐𝑔𝑔 =
𝑑𝑑𝑑𝑑 𝑚𝑚
𝜐𝜐 =
𝑝𝑝
𝑚𝑚
9.87keV/c
=
511keV/𝑐𝑐 2
= 0.0193c
= 1.93 × 10−2 c