In the previous section 2.11, we completed a discussion on Bohr model and hydrogen spectrum. We also saw the limitations of the Bohr model. It is clear that, the Bohr model does not give the complete picture of the internal structure of the atom. So scientists of that time began to think about other possibilities
• Two important developments that took place at that time were:
(i) The discovery of ‘dual behaviour of matter’ by French scientist Louis de Broglie
(ii) The discovery of ‘uncertainty’ by German scientist Werner Heisenberg
• These two discoveries laid a strong foundation upon which the further research on ‘structure of atom’ was built
• We will now see the basics about the discoveries
1. We have already seen the dual nature of electromagnetic radiations. There we saw that:
(i) Since the radiations have wavelengths, they are wave like
(ii) Since the radiations are emitted as distinct photons, they are particle like
• In some situations, the radiations exhibit wave nature
• In some other situations, radiations exhibit particle nature
2. In 1924, de Broglie found out that, matter also exhibit dual nature. That is:
• In some situations, matter exhibit wave nature
• In some other situations, matter exhibit particle nature
3. de Broglie found out that, when a particle of mass m, move with a velocity v, that particle will behave like a wave
• Wavelength ($\mathbf\small{\lambda}$) of that wave is given by: $\mathbf\small{\lambda=\frac{h}{mv}}$
♦ Where h is the Planck's constant
4. Note that, mv is equal to the momentum (p) of the particle. So we can write the wavelength in terms of momentum also: $\mathbf\small{\lambda=\frac{h}{p}}$
5. We see that, the mass m is in the denominator. So, when mass increases, the wavelength decreases
• The numerator 'h' itself is very small
• So the wavelengths of ordinary objects like automobiles will be very small
• That is why we do not feel the wave nature of ordinary objects
■ Conversely, if the mass is very small (as in the case of electron), the wavelength becomes detectable
• The findings made by de Broglie were proved experimentally:
♦ A beam of electrons was used for the experiment
♦ It was found that, the beam undergoes diffraction
♦ Diffraction is a phenomenon which is exhibited by waves
• In fact, the wave nature of the electron beam is used in the making of electron microscopes. Such microscopes can achieve a magnification of about 15 million times
7. Electrons have very small mass. So their wave nature becomes detectable
• The solved examples given at the end of this section will help us to appreciate the two items:
(i) Wave nature of macroscopic particles goes unnoticed
(Macroscopic particles are those particles which can be seen with naked eyes, with out the help of magnifying instruments)
(ii) Wave nature of microscopic particles become detectable
(Microscopic particles are those particles which cannot be seen with naked eyes. We will need magnifying instruments to see them)
1. Before discussing about ‘uncertainty’, we will first see what ‘certainty’ is:
(i) Consider a particle in motion
(ii) Consider the instant at which the reading in the stop watch is t1
♦ Let at that instant, the force acting on the particle be F1
♦ Let at that instant, the velocity of the particle be v1
♦ Let at that instant, the position of the particle be (x1,y1)
(iii) Consider the instant at which the reading in the stop watch is (t1+t)
• That is., a time interval of ‘t’ seconds has passed since the instant mentioned in (ii)
(iv) We want to know the following two items:
• The velocity of the particle at the instant (t1+t)
• The position of the particle at the instant (t1+t)
(v) If we know the three items in (ii) precisely, we can easily calculate the two items in (iv)
In fact, if we know the three items in (ii) precisely, we can fix up the 'path' that the particle will take
• So there is 'certainty' in the velocity and position of the particle
2. But this certainty is available only in the case of macroscopic particles
■ In the case of microscopic particles like electrons, the certainty is not available
(i) Let us see an example:
• Consider an electron in motion
• We want to know the items in (iv)
For that, we must first know the items in (ii)
(ii) Let us see how it can be done:
To find the position of an electron, we must first view it
• To view the electron, we must illuminate it with light
• The light reflects from the electron and falls in our eyes. When this happens, we will be able to see the electron
• Remember that, the electron is in motion. But even then we can view it and find it’s position at any instant. All we need is to ‘use a suitable light’ and precision instruments
(iv) But at the instant at which we locate the position, the photons from the ‘light used for illumination’ are hitting the electron
• In other words, the photons are colliding with the electron
• Due to this collision, the velocity of the electron will change
■ The net result is this:
♦ The location is determined with a high degree of accuracy
♦ But velocity cannot be determined with such a high degree of accuracy
(v) So what if we want to determine the velocity with a high degree of accuracy?
• We must use a dim light so that the photons will not cause appreciable change of velocity
• But then, due to the dimness, we will not be able to determine the position with a high degree of accuracy
■ The net result is this:
♦ The velocity is determined with a high degree of accuracy
♦ But location cannot be determined with such a high degree of accuracy
■ So we see that, in the case of microscopic particles, there is uncertainty
3. Let us denote the 'observed location' as x
• But due to the uncertainty, the actual location will be different from x
• We can write:
Actual location = (x ± Δx)
♦ Where Δx is the uncertainty in position
4. Similarly, let us denote the 'observed velocity' as v
• But due to the uncertainty, the actual velocity will be different from v
• We can write:
Actual velocity = (v ± Δv)
♦ Where Δv is the uncertainty in velocity
5. Now we can write the 'uncertainty' in a mathematical form:
• When Δx increases, Δv decreases
• When Δv increases, Δx decreases
6. We know that, momentum (p) = mv
So the 'uncertainty in momentum (Δp)' will be equal to mΔv
7. So we can write:
• When Δx increases, Δp decreases
• When Δp increases, Δx decreases
8. Heisenberg found out that, the product of the uncertainties in location and momentum will always be greater than or equal to $\mathbf\small{\frac{h}{4\pi}}$
That is: $\mathbf\small{\Delta x\;\times\;\Delta p\geq \frac{h}{4\pi}}$
$\mathbf\small{\Rightarrow \Delta x\;\times\;\Delta (mv)\geq \frac{h}{4\pi}}$
$\mathbf\small{\Rightarrow \Delta x\;\times\;m(\Delta v)\geq \frac{h}{4\pi}}$
$\mathbf\small{\Rightarrow \Delta x\;\times\;\Delta v\geq \frac{h}{4\pi m}}$
9. Heisenberg’s uncertainty principle states that it is impossible to determine simultaneously, the exact position and exact momentum (or velocity) of an electron
1. We have seen Bohr model of atom in the previous section
• According to that model, electron moves around the nucleus in fixed circular orbits
• That means, the electron has it’s own path
2. Now we see uncertainty principle which states that, both position and velocity cannot be determined with the same accuracy
3. If both position and velocity cannot be determined with the same accuracy, we cannot guarantee that, the electron will follow a fixed path
• So the calculations made by Bohr were not exact
4. Also note that, the Bohr model does not take the 'wave nature of electron' into account
• But it is worth to mention two points related to velocity:
♦ The velocity increases with Z
♦ The velocity decreases with n
Solved examples 2.54 to 2.64
• Two important developments that took place at that time were:
(i) The discovery of ‘dual behaviour of matter’ by French scientist Louis de Broglie
(ii) The discovery of ‘uncertainty’ by German scientist Werner Heisenberg
• These two discoveries laid a strong foundation upon which the further research on ‘structure of atom’ was built
• We will now see the basics about the discoveries
Dual Behavior of Matter
1. We have already seen the dual nature of electromagnetic radiations. There we saw that:
(i) Since the radiations have wavelengths, they are wave like
(ii) Since the radiations are emitted as distinct photons, they are particle like
• In some situations, the radiations exhibit wave nature
• In some other situations, radiations exhibit particle nature
2. In 1924, de Broglie found out that, matter also exhibit dual nature. That is:
• In some situations, matter exhibit wave nature
• In some other situations, matter exhibit particle nature
3. de Broglie found out that, when a particle of mass m, move with a velocity v, that particle will behave like a wave
• Wavelength ($\mathbf\small{\lambda}$) of that wave is given by: $\mathbf\small{\lambda=\frac{h}{mv}}$
♦ Where h is the Planck's constant
4. Note that, mv is equal to the momentum (p) of the particle. So we can write the wavelength in terms of momentum also: $\mathbf\small{\lambda=\frac{h}{p}}$
5. We see that, the mass m is in the denominator. So, when mass increases, the wavelength decreases
• The numerator 'h' itself is very small
• So the wavelengths of ordinary objects like automobiles will be very small
• That is why we do not feel the wave nature of ordinary objects
■ Conversely, if the mass is very small (as in the case of electron), the wavelength becomes detectable
• The findings made by de Broglie were proved experimentally:
♦ A beam of electrons was used for the experiment
♦ It was found that, the beam undergoes diffraction
♦ Diffraction is a phenomenon which is exhibited by waves
• In fact, the wave nature of the electron beam is used in the making of electron microscopes. Such microscopes can achieve a magnification of about 15 million times
7. Electrons have very small mass. So their wave nature becomes detectable
• The solved examples given at the end of this section will help us to appreciate the two items:
(i) Wave nature of macroscopic particles goes unnoticed
(Macroscopic particles are those particles which can be seen with naked eyes, with out the help of magnifying instruments)
(ii) Wave nature of microscopic particles become detectable
(Microscopic particles are those particles which cannot be seen with naked eyes. We will need magnifying instruments to see them)
Heisenberg’s Uncertainty Principle
1. Before discussing about ‘uncertainty’, we will first see what ‘certainty’ is:
(i) Consider a particle in motion
(ii) Consider the instant at which the reading in the stop watch is t1
♦ Let at that instant, the force acting on the particle be F1
♦ Let at that instant, the velocity of the particle be v1
♦ Let at that instant, the position of the particle be (x1,y1)
(iii) Consider the instant at which the reading in the stop watch is (t1+t)
• That is., a time interval of ‘t’ seconds has passed since the instant mentioned in (ii)
(iv) We want to know the following two items:
• The velocity of the particle at the instant (t1+t)
• The position of the particle at the instant (t1+t)
(v) If we know the three items in (ii) precisely, we can easily calculate the two items in (iv)
In fact, if we know the three items in (ii) precisely, we can fix up the 'path' that the particle will take
• So there is 'certainty' in the velocity and position of the particle
2. But this certainty is available only in the case of macroscopic particles
■ In the case of microscopic particles like electrons, the certainty is not available
(i) Let us see an example:
• Consider an electron in motion
• We want to know the items in (iv)
For that, we must first know the items in (ii)
(ii) Let us see how it can be done:
To find the position of an electron, we must first view it
• To view the electron, we must illuminate it with light
• The light reflects from the electron and falls in our eyes. When this happens, we will be able to see the electron
• Remember that, the electron is in motion. But even then we can view it and find it’s position at any instant. All we need is to ‘use a suitable light’ and precision instruments
(iv) But at the instant at which we locate the position, the photons from the ‘light used for illumination’ are hitting the electron
• In other words, the photons are colliding with the electron
• Due to this collision, the velocity of the electron will change
■ The net result is this:
♦ The location is determined with a high degree of accuracy
♦ But velocity cannot be determined with such a high degree of accuracy
(v) So what if we want to determine the velocity with a high degree of accuracy?
• We must use a dim light so that the photons will not cause appreciable change of velocity
• But then, due to the dimness, we will not be able to determine the position with a high degree of accuracy
■ The net result is this:
♦ The velocity is determined with a high degree of accuracy
♦ But location cannot be determined with such a high degree of accuracy
■ So we see that, in the case of microscopic particles, there is uncertainty
3. Let us denote the 'observed location' as x
• But due to the uncertainty, the actual location will be different from x
• We can write:
Actual location = (x ± Δx)
♦ Where Δx is the uncertainty in position
4. Similarly, let us denote the 'observed velocity' as v
• But due to the uncertainty, the actual velocity will be different from v
• We can write:
Actual velocity = (v ± Δv)
♦ Where Δv is the uncertainty in velocity
5. Now we can write the 'uncertainty' in a mathematical form:
• When Δx increases, Δv decreases
• When Δv increases, Δx decreases
6. We know that, momentum (p) = mv
So the 'uncertainty in momentum (Δp)' will be equal to mΔv
7. So we can write:
• When Δx increases, Δp decreases
• When Δp increases, Δx decreases
8. Heisenberg found out that, the product of the uncertainties in location and momentum will always be greater than or equal to $\mathbf\small{\frac{h}{4\pi}}$
That is: $\mathbf\small{\Delta x\;\times\;\Delta p\geq \frac{h}{4\pi}}$
$\mathbf\small{\Rightarrow \Delta x\;\times\;\Delta (mv)\geq \frac{h}{4\pi}}$
$\mathbf\small{\Rightarrow \Delta x\;\times\;m(\Delta v)\geq \frac{h}{4\pi}}$
$\mathbf\small{\Rightarrow \Delta x\;\times\;\Delta v\geq \frac{h}{4\pi m}}$
9. Heisenberg’s uncertainty principle states that it is impossible to determine simultaneously, the exact position and exact momentum (or velocity) of an electron
Significance of uncertainty principle
1. We have seen Bohr model of atom in the previous section
• According to that model, electron moves around the nucleus in fixed circular orbits
• That means, the electron has it’s own path
2. Now we see uncertainty principle which states that, both position and velocity cannot be determined with the same accuracy
3. If both position and velocity cannot be determined with the same accuracy, we cannot guarantee that, the electron will follow a fixed path
• So the calculations made by Bohr were not exact
4. Also note that, the Bohr model does not take the 'wave nature of electron' into account
• But it is worth to mention two points related to velocity:
♦ The velocity increases with Z
♦ The velocity decreases with n
The link given below can be used to see some solved examples presented in pdf format
Solved examples 2.54 to 2.64
This completes our present discussion on the 'dual nature of matter' and 'uncertainty principle'. These discoveries paved way for the development of the 'quantum mechanical model of atom'. We will see it in the next section
