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Module- 1
Quantum Mechanics
Wave Nature of Particles (Wave-particle dualism)
From the phenomenon of interference and diffraction light is considered purely as waves.
From Planck’s idea of quantization Einstein proposed that light consists of discrete units of energy
known as photons, which was later confirmed by the photoelectric effect experiment. Another
effect that revealed the quantized nature of radiation was the elastic scattering of light on particles,
called Compton Effect or Compton Scattering. Because of such dual nature observed of radiation
and light, Louis de Broglie of France in 1924 put forward a hypothesis that
“Nature loves symmetry, if the radiation behaves as particle under certain circumstances and as
waves under certain other circumstances, then one can even expect that, entities which ordinarily
behave as particles to exhibit properties attributable to only waves under appropriate
circumstances.”
De-Broglie extended the wave-particle duality of light to the material particles.
If a light wave can act as a wave and as a particle at other times, then particles such as electrons
also act as waves at times. This is known as de-Broglie hypothesis. According to de Broglie
hypothesis “Every moving particle has a wave associated with it.” The waves associated with the
particles are known as de-Broglie waves or matter waves.
De-Broglie wave length of matter waves
A particle with mass “m” moving with velocity “c” possess energy given by
2
…………………(1) (Einstein’s equation)
From Planck’s theory of quantization
ℎ𝑐
𝜆
From equation (1) and (2)
2
ℎ𝑐
𝜆
Or, 𝜆 =
ℎ
𝑚𝑐
Since 𝑝 = 𝑚𝑣 = 𝑚𝑐
From equation (3)
The relation 𝜆 =
ℎ
𝑝
is known as de-Broglie equation and the wavelength 𝜆 is called the de-Broglie
wavelength. The waves associated with the moving particle are called matter waves or de Broglie
waves.
De-Broglie wavelength of accelerated electron
If a charged particle, say an electron is accelerated by a potential difference of V volts, then its
kinetic energy is given by K.E. = eV
2
If p be the momentum of the electron, then,
Squaring both sides, we have
2
2
2
From equation (3)
2
2
Or,
2
From de Broglie’s hypothesis we know that
Since h, m and e are universal physical constants. Substituting the value of the constants in
equation (5)
[
6. 626 × 10
− 34
√ 2 ( 9. 11 × 10
− 31
)( 1. 602 × 10
− 19
]
1. 226 × 10
− 9
One of the important consequences of the uncertainty principle is seen in the broadening
of spectral lines. Atoms have electrons that can be excited to higher energy levels and electrons in
atoms like to stay in the lowest possible energy state (ground state). If an electron is pushed to a
higher energy level (excited state), it is unstabl e. After some time, the electron will naturally fall
back to a lower level. When it does, it releases energy in the form of a photon (light). This process
is called spontaneous emission. Because the excited state is unstable, the electron cannot stay there
forever. The average time the electron spends in the excited state is called its lifetime (Δt). For
most atoms, this is typically nanoseconds to microseconds. Since the state exists only for a short
time, the exact energy of the photon emitted cannot be perfectly sharp. There is an uncertainty in
energy of emitted photon ∆𝐸. This uncertainty in energy translates to an uncertainty in frequency
Instead of one sharp frequency, the atom emits light over a range of nearby frequencies. The
spectral line therefore has a spread in frequency and this spread is called the linewidth or
broadening.
Now
Differentiating the above equation with respect to 𝜆
ℎ𝑐
𝜆
2
From uncertainty principle
∆𝐸 × ∆𝑡 =
Or,
ℎ
4 𝜋×∆𝑡
Substituting ∆𝐸 from equation ( 3 ) to equation ( 4 )
2
4 𝜋 × ∆𝑡
Or,
2
Neglecting the negative sign, the wavelength spread is given by
2
The above equation shows that for a finite lifetime of
the excited state, the measured value of the emitted
photon wavelength will have spread of wavelengths
around the mean value 𝜆. As a result the spectral lines
are never infinitely sharp, they exhibit a natural width.
The shorter the lifetime of the excited state, the greater
the energy uncertainty and hence the broader the
spectral line.
Principle of complementarity:
Principle of complementarity was proposed by Niels Bohr in 1928. It deals with the wave–
particle duality of matter and radiation. Bohr’s complementarity principle says that “A quantum
system can show either wave-like behaviour or particle-like behaviour, depending on the type of
experiment performed, but never both at the same time.”
Wave-like behavior and particle-like behaviour of quantum objects (such as electrons and photons)
are two complementary aspects of the same physical reality. Both aspects are necessary for a full
description of quantum phenomena, but they cannot be observed simultaneously in the same
experiment.
Wave function and probability interpretation
Waves represent the propagation of disturbance in a medium. We are familiar with sound
waves, light waves, and water waves. These waves are characterized by some quantity that varies
with position and time. Light waves consist of variations of electric and magnetic fields in space,
and sound waves consist of pressure variations. Similarly, the microscopic particles (for example
electron, proton, neutron etc.) exhibit wave properties, it is assumed that a quantity ψ represents
a de-Broglie wave. This quantity is called a wave function ψ. Ψ describes the wave as a function
of position and time. However, it has no direct physical significance, as ψ is a complex quantity.
For any particle of mass “m” moving along the x-axis; at any position and momentum time the
behaviour and motion of the particle is given by the wave function 𝛹
The wave function can be represented by
𝑖(𝑘𝑥−𝜔𝑡)
Max Born Interpretation of wave function
Schrodinger equation
As per de-Broglie hypothesis every moving particle has a wave associated with it, which is also
known as matter wave. Further, Erwin Schrödinger in continuation to de-Broglie’s hypothesis
introduced a differential wave equation of second order to explain the wave nature of matter and
particle associated to wave. Schrodinger equation plays the same role in Q.M as Newton’s laws in
classical mechanics.
The Schrodinger equation can be set up in two contexts
❖ Time Dependent Schrodinger equation in which position and time variation of the wave
function. It involves the imaginary quantity “i”.
❖ Time Independent Schrodinger equation in which the wave function can have variation only
with position but not with time. It does not involve “i”.
Time Independent Schrodinger equation
Consider a particle is moving along x-axis and exhibiting the simple harmonic wave pattern
represented by
𝑑
2
𝑦
𝑑𝑥
2
1
𝑣
2
𝑑
2
𝑦
𝑑𝑡
2
The solution of equation (1) is
𝑖(𝑘𝑥−𝜔𝑡)
where y is the displacement, ω is the angular frequency, A is the amplitude and v is the velocity
of the wave.
The matter waves associated with the matter behaves same as mentioned in the equation (1), then
equation ( 2 ) can be written as
2
2
2
2
2
where v is the velocity of the matter waves and 𝜓 is the wave function.
The solution for the equation ( 4 ) can be written as
𝑖(𝑘𝑥−𝜔𝑡)
Where 𝜓
0
is the amplitude of the matter wave.
Differentiating equation ( 5 ) with respect to “t”
𝑖(𝑘𝑥−𝜔𝑡)
Again differentiating the above equation
2
2
2
2
−𝑖
( 𝑘𝑥−𝜔𝑡
)
2
𝑖
( 𝑘𝑥−𝜔𝑡
)
2
2
2
Substituting equation ( 6 ) to equation ( 4 ), we get
2
2
2
2
2
2
2
2
If 𝜆 𝑎𝑛𝑑 𝜈 are the wavelength and frequency of the wave, then
𝜔 = 2 𝜋𝜈 and 𝑣 = 𝜈𝜆
Substituting 𝜔 𝑎𝑛𝑑 𝑣, equation ( 6 ) becomes
𝑑
2
𝜓
𝑑𝑥
2
4 𝜋
2
𝜆
2
Substituting 𝜆 =
in the above equation, we get
2
2
2
2
2
The total energy of the particle is given by
E = Kinetic Energy +Potential Energy
2
If p is the momentum of the particle along X-direction then p=mv, the above
equation may be written as
2
2
Substituting the value of 𝑝
2
𝑖𝑛 the equation ( 8 ), we get
∗
∞
−∞
Expectation value of energy (
Operator for energy is the Hamiltonian:
2
∗
∞
−∞
Physical Significance of expectation values:
Expectation values are quantum averages, what we expect if the same experiment is
repeated many times. They bridge the gap between quantum mechanics (probabilistic) and
classical mechanics (deterministic).
Examples:
〈𝑥〉 𝑔𝑖𝑣𝑒𝑠 the “center of mass” of the probability distribution, 〈𝑝〉 gives the average motion of the
particle and 〈𝐸〉 gives the mean energy of the system.
Particle in one-dimensional infinitely deep potential well (particle in 1D box)
Let us consider a particle of mass “m” confined to a one-dimensional rigid box of width
“a”. It can move freely within the region 0 < 𝑥 < 𝑎 but subject to strong forces at x = 0 and x =
a. Therefore, it can never cross to the right of the region 𝑥 > 𝑎 or to the left of 0. It means that
V(x) = 0 in the region 0 < 𝑥 < 𝑎 𝑎𝑛𝑑 𝑟𝑖𝑠𝑒𝑠 𝑡𝑜 infinity (𝑉(𝑥) = ∞) at x=0 and x=a. This situation
is called a one-dimensional potential box.
In terms of boundary conditions imposed by the problem, the potential function is given as
V(x) =0; for 0< x< a
V(x) = ∞ 𝑓𝑜𝑟 𝑥 ≥ 𝑎 𝑎𝑛𝑑 𝑥 ≤ 0
Inside the well, the Schrodinger equation is given by
𝑑
2
𝜓
𝑑𝑥
2
8 𝜋
2
𝑚
ℎ
2
E𝜓 = 0 , (V=0) − − − − − − ( 1 )
Let
8 𝜋
2
𝑚
ℎ
2
E = 𝐾
2
Substituting equation (2) in equation ( 1 )
2
2
2
The solution of the equation ( 3 ) is given by
𝑖𝐾𝑥
−𝑖𝐾𝑥
Boundary value
𝜓 = 0 𝑎𝑡 𝑥 = 0 ………….. Condition I
𝜓 = 0 𝑥 = 𝑎 ………….. Condition II
Substituting the condition I in equation ( 4 )
0
0
or, A=-B − − − − − − − ( 5 )
Substituting condition II in equation ( 4 )
𝑖𝐾𝑎
−𝑖𝐾𝑎
A(cosKa+isinKa)+B(cosKa-isinKa)=
Using equation ( 5 )
A(cosKa+isinKa)-A(cosKa-isinKa)=
2iAsinKa = 0
Since 2i𝐴 ≠ 0 , 𝑠𝑖𝑛𝐾𝑎 = 0
Or, Ka = n𝜋
K =
𝑛𝜋
𝑎
where n = 0,1,2,3, ……..
n is called quantum number which is either zero or a positive integer
Substituting equation ( 5 ) and equation ( 6 ) in equation ( 4 )
𝑛
2
[
𝑎
0
𝑎
𝑜
] = 1
𝐶
2
2
[𝑎 −
𝑎
2 𝑛𝜋
sin ( 2 𝑛𝜋)] = 1
𝐶
2
𝑎
2
= 1 (sin 2 𝑛𝜋 = 0)
Thus the normalized wave function of a particle in an one-dimensional potential well is given by
𝑛
2
𝑎
𝑛𝜋
𝑎
Eigen functions, Probability densities and Energy Eigenvalues for particle in an Infinite
Potential Well
Since the particle in an infinite potential well is a problem under quantum mechanical
conditions, the prime questions to be considered are the most probable location of the particle in
the well and its energies, both to be evaluated for the different permitted states.
Eigen function for particle in infinite potential well
𝑛
2
𝑎
𝑛𝜋
𝑎
x …………. (10)
We can write the eigenfunctions 𝜓 1
2
3
, … …. for particle in the well by putting n =1, 2,
3,………. respectively in the equation
Let us consider the first 3 cases
Case (i), n = 1:
This is the ground state and the particle is found in this state.
For n =1, the eigen function is
1
From equation ( 11 ), 𝜓 1
= 0 when 𝑠𝑖𝑛 (
𝜋
𝑎
𝜋
𝑎
) 𝑥 = sin𝑚𝜋; m = 0, 1, 2, 3, …….
For m = 0; x=0 and m =1; x = a
So 𝜓
1
= 0 for both x = 0 and x = a
From equation ( 11 ), 𝜓 1
has maximum value when 𝑠𝑖𝑛 (
𝜋
𝑎
) 𝑥 has maximum value
𝜋
𝑎
𝑘𝜋
2
); k =1,3,5,7,……..
For k = 1; 𝑠𝑖𝑛 (
𝜋
𝑎
𝜋
2
𝑎
2
1
has maximum value
Regarding the energy of the particle, using equation ( 8 ), the energy in the ground state by putting
n = 1.
1
ℎ
2
8 𝑚𝑎
2
= E
0
This is the energy eigen value for the ground state.
Case (ii), n = 2
This is the first excited state i.e., the next immediate higher state permitted for the particle after
the ground state.
From equation ( 10 ), the eigenfunction for this state is,
2
Now, 𝜓
2
= 0 when 𝑠𝑖𝑛 (
2 𝜋
𝑎
2 𝜋
𝑎
) 𝑥 = sin 𝑚𝜋; 𝑚 = 0 ,1,2,3,..
For m=0, x =0; m=1, x=a/2 and m=2, x=a
2
= 0 at x = 0, a/2 and a
2
𝑖𝑠 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 when 𝑠𝑖𝑛 (
2 𝜋
𝑎
) 𝑥 is maximum
2 𝜋
𝑎
) 𝑥 = sin k𝜋/ 2 ; 𝑘 =1,3,5,..
For k=1, x =a/4 and k=3, x=3a/
2
is maximum at x = a/4, 3a/4.
Further from equation ( 8 ), for the first excited states; n=
2
ℎ
2
8 𝑚𝑎
2
) represents the energy eigen value for the 2
nd
excited state
2
0
Thus the energy in the first excited state is 4 times the zero point energy.
Case (iii), n = 3:
Extension to 2D and 3D (Qualitative):
Now confine the particle inside a square box of side a (walls in both x- and y-directions). In
two-dimensional potential box, the particle (electron) can move in both “x” and “y” directions,
therefore instead of one quantum number ‘n’, two quantum numbers 𝑛
𝑥
and 𝑛
𝑦
are required, each
taking values 1, 2, 3, ………….. Let us consider a particle enclosed in 2-D potential well of length
“a” and “b” along X- and Y- axis respectively as shown in the figure.
Since the particle inside 2-D well has an elastic collision with the walls
The potential energy of the particle inside the well is constant and can be
taken as zero for simplicity. Outside and on the walls of the well, the
potential energy is infinity.
The allowed energies of the particle are given by:
𝑛
𝑥,
𝑛
𝑦
2
2
𝑥
2
𝑦
2
The total energy comes from contributions in both directions. Degeneracy occurs in this case,
because different combinations of 𝑛
𝑥
and 𝑛
𝑦
can produce the same energy. For example, the states
𝑥
𝑦
=1) and (𝑛
𝑥
𝑦
=2) have the same energy.
Three-Dimensional (3D) Infinite Potential Well:
In three dimensions, the particle is confined inside a cube of side a. The motion of the
particle is restricted along the x-, y-, and z-directions. In this case, three quantum numbers 𝑛
𝑥
𝑦
and 𝑛 𝑧
are required, each taking values 1, 2, 3, ……..The energy depends on the sum of the squares
of all three quantum numbers. The allowed energies of the particle are given by:
𝑛
𝑥,
𝑛
𝑦
,𝑛
𝑧
2
2
𝑥
2
𝑦
2
𝑧
2
More degeneracy occurs in three dimensions because different sets of quantum numbers can give
the same energy. For example, the states (𝑛 𝑥
𝑦
𝑧
=1) and (𝑛
𝑥
𝑦
𝑧
=1) and 𝑛
𝑥
𝑦
𝑧
=2) all have the same energy. The wavefunctions represent three-dimensional standing
waves inside the cube.
Finite Potential Well and Tunneling:
The finite square well is one of the cornerstone problems in non-relativistic quantum
mechanics. Unlike the infinite square well, where the particle is strictly confined, the finite well
allows the wave function to penetrate beyond the well boundaries due to the phenomenon of
quantum tunneling. This model provides important insights into real systems such as electrons in
quantum dots, nuclei, and semiconductor. Figure shows a potential well with square corners that
is U high and L wide and contains a particle whose energy E is less than U. According to classical
mechanics, when the particle strikes the sides of the well, it bounces off without entering regions
I and III. In quantum mechanics, the particle also bounces back and forth, but now it has certain
probability of penetrating into regions I and II even though E<U.
The Schrödinger’s wave equation outside the box in region I and III can be given as
2
2
2
2
Which we can write in the more convenient form
𝑑
2
𝜓
𝑑𝑥
2
2
𝜓 = 0 𝑥 < 0 and 𝑥 > 𝐿
Where 𝑘
2
8 𝜋
2
𝑚(𝑈−𝐸)
ℎ
2
The solution to equation (1) can be written as
𝐼
𝑘
1
𝑥
−𝑘
1
𝑥
And
𝐼𝐼𝐼
𝑘
1
𝑥
−𝑘
1
𝑥
Both 𝜓 𝐼
and 𝜓
𝐼𝐼𝐼
must be finite everywhere. Since 𝑥 → −∞, 𝑒
−𝑘
1
𝑥
→ ∞, so in equation (1), D
must be zero
and 𝑥 → ∞, 𝑒
𝑘 1
𝑥
→ ∞ so in equation (2), F must be zero
−𝑘
1
𝐿
𝑘
1
𝐶
𝑘
2
2
2
Now applying the boundary conditions 8 d in equations ( 5 ) and (7)
2
2
2
2
1
−𝑘 1
𝐿
Substitute value of A and B from equation (9) and (10) in the above equation
2
𝑘
1
𝐶
𝑘
2
2
2
2
1
−𝑘 1
𝐿
Or, - C 𝑐𝑜𝑠𝑘
2
𝑘
2
𝐶
𝑘
1
2
−𝑘
1
𝐿
Substitute value of 𝐺𝑒
−𝑘 1
𝐿
from equation (11) in equation (12)
- C 𝑐𝑜𝑠𝑘
2
𝑘
2
𝐶
𝑘
1
2
𝑘
1
𝐶
𝑘
2
2
2
or,
2
𝑘
2
𝑘 1
2
𝑘
1
𝑘 2
2
2
Divide by 𝑠𝑖𝑛𝑘 2
𝐿 in the above equation
2
𝑘
2
𝑘
1
𝑘
1
𝑘
2
2
2
𝑘
2
𝑘
1
𝑘
1
𝑘
2
As discussed above, 𝑘 2
8 𝜋
2
𝑚𝐸
ℎ
2
and 𝑘
1
8 𝜋
2
𝑚(𝑈−𝐸)
ℎ
2
Substituting values of 𝑘
1
and 𝑘
2
in equation (13) and solving the equation, energy will be
quantized. The complete wavefunction and their probability densities are shown below:
Quantum Tunneling:
Classically, a particle with E<U cannot cross a barrier. Quantum mechanically, the wave function
extends into and even through the barrier, giving a finite probability of transmission called
Tunneling.
Transmission probability
−𝑘
2
𝐿
Where 𝑘 2
√
2 𝑚(𝑈−𝐸)
ℏ
Physical Examples of tunneling:
- Alpha decay in nuclei.
- Scanning tunneling microscope (STM).
- Tunnel diodes in electronics.
- Proton tunneling in stellar fusion
