THE WAVE BEHAVIOUR OF MATTER
In the years following the development of Bohr's model for the hydrogen atom, the dual nature of
radiant energy became a familiar concept. Depending on the experimental circumstances, radiation
appears to have either a wavelike or a particle-like (photon) character. Louis de Broglie (1892-
1987), who was working on his Ph.D. thesis in physics at the Sorbonne in Paris, argued that
because electromagnetic radiation could be considered to consist of particles, called photons, yet
at the same time exhibit wave-like properties, then the same might be true of electrons. De Broglie
suggested that as the electron moves about the nucleus, it is associated with a particular
wavelength. He went on to propose that the characteristic wavelength of the electron, or of any
other particle, depends on its mass, m, and on its velocity, v, (where h is Planck's constant):
ℎ
𝜆=
𝑚𝑣
The quantity mv for any object is called its momentum.
Because de Broglie's hypothesis is applicable to all matter, any object of mass m and velocity v
would give rise to a characteristic matter wave. However, the above equation indicates that the
wavelength associated with an object of ordinary size, such as a golf ball, is so tiny as to be
completely out of the range of any possible observation.
In 1927, a German physicist Werner Heisenberg stated the uncertainty principle which is the
consequence of dual behaviour of matter and radiation. This principle states that it is impossible
to determine simultaneously, the exact position and exact momentum (or velocity) of an
electron. Mathematically, it can be given as in equation as:
ℎ ℎ ℎ
Δx x Δp ≥ 4π or Δx x Δ(mvx) ≥ 4π or Δx x Δvx ≥ 4πm
where ∆x is the uncertainty in position and ∆px ( or ∆vx) is the uncertainty in momentum (or
velocity) of the particle. If the position of the electron is known with high degree of accuracy (∆x
is small), then the velocity of the electron will be uncertain [∆(vx) is large]. On the other hand, if
the velocity of the electron is known precisely (∆(vx ) is small), then the position of the electron
will be uncertain (∆x will be large).
The effect of Heisenberg Uncertainty Principle is significant only for motion of microscopic
objects and is negligible for that of macroscopic objects. Consider the following examples:
Page 1 of 6
, i. If uncertainty principle is applied to an object of mass, say about a milligram (10–6 kg),
6.626 ×10−34 𝐽𝑆
Δx . Δv = h/4πm = 4𝑥3.14 ×10−6 𝑘𝑔 ≈ 10−28 m2s-1
The value of ∆v∆x obtained is extremely small and is insignificant. Therefore, one may
say that in dealing with milligram-sized or heavier objects, the associated uncertainties are
hardly of any real consequence.
ii. For an electron whose mass is 9.11×10–31 kg.,
6.626 ×10−34 𝐽𝑆
Δx . Δv = h/4πm = 4𝑥3.14 × 𝑥 9.11 𝑥 10−31 𝑘𝑔 ≈ 10−4 m2s-1
It, therefore, means that if one tries to find the exact location of the electron, say to an
uncertainty of only 10–8 m, then the uncertainty ∆v in velocity would be
10−4 𝑚2 𝑠−1
≈ 104 ms-1
10−4 𝑚
which is so large that it rules out the existence of definite paths or trajectories of
electrons and other similar particles.
De Broglie's hypothesis and Heisenberg's uncertainty principle set the stage for a new and more
broadly applicable theory of atomic structure. In this new approach, any attempt to define
precisely the instantaneous location and momentum of the electron is abandoned. The wave nature
of the electron is recognized, and its behavior is described in terms appropriate to waves. The
result is a model that precisely describes the energy of the electron while describing its location
not precisely, but in terms of probabilities.
The branch of science that takes into account this dual behaviour of matter is called quantum
mechanics. Quantum mechanics is a theoretical science that deals with the study of the motions
of the microscopic objects that have both observable wave like and particle like properties.
Quantum mechanics was developed independently in 1926 by Werner Heisenberg and an
Austrian physicist Erwin Schrödinger (1887-1961). In SCH 101, however, we shall discuss the
quantum mechanics that relate to the ideas of wave motion. The fundamental equation of quantum
mechanics was developed by Schrödinger and it won him the Nobel Prize in Physics in 1933. This
equation which incorporates wave particle duality of matter as proposed by de Broglie is quite
complex and knowledge of higher mathematics is needed to solve it. Schrodinger introduced the
wavefunction, ψ (psi), a mathematical function of the position coordinates x, y, and z of an
Page 2 of 6
In the years following the development of Bohr's model for the hydrogen atom, the dual nature of
radiant energy became a familiar concept. Depending on the experimental circumstances, radiation
appears to have either a wavelike or a particle-like (photon) character. Louis de Broglie (1892-
1987), who was working on his Ph.D. thesis in physics at the Sorbonne in Paris, argued that
because electromagnetic radiation could be considered to consist of particles, called photons, yet
at the same time exhibit wave-like properties, then the same might be true of electrons. De Broglie
suggested that as the electron moves about the nucleus, it is associated with a particular
wavelength. He went on to propose that the characteristic wavelength of the electron, or of any
other particle, depends on its mass, m, and on its velocity, v, (where h is Planck's constant):
ℎ
𝜆=
𝑚𝑣
The quantity mv for any object is called its momentum.
Because de Broglie's hypothesis is applicable to all matter, any object of mass m and velocity v
would give rise to a characteristic matter wave. However, the above equation indicates that the
wavelength associated with an object of ordinary size, such as a golf ball, is so tiny as to be
completely out of the range of any possible observation.
In 1927, a German physicist Werner Heisenberg stated the uncertainty principle which is the
consequence of dual behaviour of matter and radiation. This principle states that it is impossible
to determine simultaneously, the exact position and exact momentum (or velocity) of an
electron. Mathematically, it can be given as in equation as:
ℎ ℎ ℎ
Δx x Δp ≥ 4π or Δx x Δ(mvx) ≥ 4π or Δx x Δvx ≥ 4πm
where ∆x is the uncertainty in position and ∆px ( or ∆vx) is the uncertainty in momentum (or
velocity) of the particle. If the position of the electron is known with high degree of accuracy (∆x
is small), then the velocity of the electron will be uncertain [∆(vx) is large]. On the other hand, if
the velocity of the electron is known precisely (∆(vx ) is small), then the position of the electron
will be uncertain (∆x will be large).
The effect of Heisenberg Uncertainty Principle is significant only for motion of microscopic
objects and is negligible for that of macroscopic objects. Consider the following examples:
Page 1 of 6
, i. If uncertainty principle is applied to an object of mass, say about a milligram (10–6 kg),
6.626 ×10−34 𝐽𝑆
Δx . Δv = h/4πm = 4𝑥3.14 ×10−6 𝑘𝑔 ≈ 10−28 m2s-1
The value of ∆v∆x obtained is extremely small and is insignificant. Therefore, one may
say that in dealing with milligram-sized or heavier objects, the associated uncertainties are
hardly of any real consequence.
ii. For an electron whose mass is 9.11×10–31 kg.,
6.626 ×10−34 𝐽𝑆
Δx . Δv = h/4πm = 4𝑥3.14 × 𝑥 9.11 𝑥 10−31 𝑘𝑔 ≈ 10−4 m2s-1
It, therefore, means that if one tries to find the exact location of the electron, say to an
uncertainty of only 10–8 m, then the uncertainty ∆v in velocity would be
10−4 𝑚2 𝑠−1
≈ 104 ms-1
10−4 𝑚
which is so large that it rules out the existence of definite paths or trajectories of
electrons and other similar particles.
De Broglie's hypothesis and Heisenberg's uncertainty principle set the stage for a new and more
broadly applicable theory of atomic structure. In this new approach, any attempt to define
precisely the instantaneous location and momentum of the electron is abandoned. The wave nature
of the electron is recognized, and its behavior is described in terms appropriate to waves. The
result is a model that precisely describes the energy of the electron while describing its location
not precisely, but in terms of probabilities.
The branch of science that takes into account this dual behaviour of matter is called quantum
mechanics. Quantum mechanics is a theoretical science that deals with the study of the motions
of the microscopic objects that have both observable wave like and particle like properties.
Quantum mechanics was developed independently in 1926 by Werner Heisenberg and an
Austrian physicist Erwin Schrödinger (1887-1961). In SCH 101, however, we shall discuss the
quantum mechanics that relate to the ideas of wave motion. The fundamental equation of quantum
mechanics was developed by Schrödinger and it won him the Nobel Prize in Physics in 1933. This
equation which incorporates wave particle duality of matter as proposed by de Broglie is quite
complex and knowledge of higher mathematics is needed to solve it. Schrodinger introduced the
wavefunction, ψ (psi), a mathematical function of the position coordinates x, y, and z of an
Page 2 of 6
