Question

Planck's constant \(h\) has the dimensions of action (energy \(x\) time or momentum \(\times\) distance) which suggests that the action of any periodic phenomenon may have to be a multiple of h. Accordingly Bohr constructed a model of the hydrogen atom in which the action of the single orbiting electron was quantized, requiring \(2 \pi r m v=n h\), \(n=1,2, \ldots\), where \(m\) is the mass of the electron, \(v\) its speed, and \(r\) the radius of the orbit. This led to a hydrogen spectrum which fitted the then known facts. Show that Bohr's hypothesis (1913) is equivalent to the assumption that a permissible orbit must contain an integral number of de Broglie electron waves.

Short answer

Answer: Bohr's hypothesis is equivalent to the assumption of orbits containing an integral number of de Broglie electron waves because both conditions lead to the same result of quantized electron orbits in a hydrogen atom. Through a series of substitutions and mathematical manipulations using Bohr's quantization condition and the de Broglie wavelength, it is shown that an orbit is permissible if its perimeter is an integer multiple of the de Broglie wavelength, which implies that the orbit contains an integral number of electron waves.

Step-by-step solution

Write down Bohr's quantization condition

Bohr's hypothesis states that the action of the orbiting electron in a hydrogen atom is quantized, which can be expressed as: $$2 \pi r m v = n h, \quad n=1,2, \ldots$$ where \(m\) is the mass of the electron, \(v\) its speed, \(r\) the radius of the orbit, and \(n\) a positive integer.

Introduce the de Broglie wavelength

According to the de Broglie hypothesis, the wavelength \(\lambda\) of a particle with mass \(m\) and velocity \(v\) is given by the expression: $$\lambda = \frac{h}{mv}$$ where \(h\) is the Planck's constant.

Write the de Broglie wavelength in terms of the orbit radius

Now, we will write the de Broglie wavelength in terms of the orbit radius, \(r\). To do this, we can first write the momentum \(mv\) as: $$mv = \frac{h}{\lambda}$$ And substitute this expression into Bohr's quantization condition: $$ 2\pi r \left(\frac{h}{\lambda}\right) = nh$$

Solve the equation for the orbit radius

Now, we will solve the above equation for the orbit radius, \(r\). We can do this by dividing both sides by \(2\pi h\): $$r = \frac{n\lambda}{2\pi}$$

Show the condition for a permissible orbit

Now, we will derive the condition for an orbit to be permissible, which is that it must contain an integral number of de Broglie electron waves. We can find the total perimeter of an orbit that contains an integral number of wavelengths by multiplying the integer number of wavelengths \(p\) by \(\lambda\): $$P=p\lambda$$ Now, let us plug the expression for \(r\) in terms of the de Broglie wavelength from Step 4: $$P = p\left(\frac{2\pi r}{n}\right)$$ We can conclude that an orbit is permissible if its perimeter is an integer multiple of the de Broglie wavelength, which means the orbit contains an integral number of de Broglie electron waves. This shows that Bohr's hypothesis is indeed equivalent to the assumption of an orbit containing an integral number of electron waves.

Key concepts

Planck's Constant

Planck's constant, denoted as \(h\), is a fundamental quantity in quantum mechanics and has the unit of action, which can be expressed as energy multiplied by time (Joule-seconds) or momentum times distance. It was first introduced by Max Planck in 1900 and is essential to describe the quantization of energy in systems at the atomic scale.

According to quantum theory, energy is not continuous, but rather 'quantized', meaning it can only exist in discrete amounts called quanta. The energy \(E\) of a photon, which is a particle of light, is related to its frequency \(f\) by the equation \(E = h f\). Planck's constant bridges the gap between the macroscopic world we observe daily and the microscopic world of atoms and particles, providing a scale at which quantum effects become significant.

Quantization of Action

The concept of quantization of action means that on the microscopic level, physical quantities such as angular momentum or the action in a system, do not take on any continuous value but are quantized in discrete 'packets'. This idea was revolutionary because it contradicted classical mechanics, which viewed physical properties as continuous ranges.

Bohr’s hypothesis applied this concept to the angular momentum of electrons in atoms, postulating that the angular momentum is quantized and can only take on certain discrete values. This was characterized by the equation \(2 \pi r m v = n h\), where \(n\) is a positive integer, \(r\) is the orbit radius, \(m\) the mass of the electron, \(v\) its speed, and \(h\) is Planck's constant. This hypothesis explained why electrons do not radiate energy and spiral into the nucleus, a major question classical physics could not address.

De Broglie Wavelength

The de Broglie hypothesis introduces a wave-particle duality where particles like electrons exhibit both wave-like and particle-like properties. Louis de Broglie proposed that any particle with momentum \(p\) has an associated wavelength called the de Broglie wavelength, given by \(\lambda = \frac{h}{p}\).

For an electron in orbit around an atom's nucleus, its de Broglie wavelength is related to its velocity and the Planck's constant. The equation \(\lambda = \frac{h}{mv}\) demonstrates this relationship and implies that even particles with mass have a wave-like nature, which was a groundbreaking idea contrasting classical notions of particles.

Hydrogen Atom Model

The Bohr model for the hydrogen atom was a pioneering model that combined classical physics with quantum hypothesis to explain the discrete energy levels observed in the hydrogen spectrum. In this model, electrons travel in fixed orbits around the nucleus with quantized angular momenta, avoiding the problem of the collapse of the atom which classical physics could not resolve.

The conditions for electron orbits were such that only certain radii and energy levels were allowed, exemplified by the equation \(2 \pi r m v = n h\). Each orbit corresponds to a specific energy level, and when an electron transitions between these levels, they absorb or emit light at particular wavelengths, leading to the characteristic spectrum of hydrogen.

Special Relativity

Special relativity, developed by Albert Einstein in 1905, fundamentally changed our understanding of space, time, and motion. It introduced the idea that the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum is the same, no matter the motion of the source or observer.

One of the key insights of special relativity is time dilation, which shows that time can pass at different rates for different observers based on their relative velocities. Another is mass-energy equivalence, encapsulated in the famous equation \(E=mc^2\), which implies that mass and energy are interchangeable. While special relativity mainly affects objects moving at speeds close to the speed of light and is not directly involved in the Bohr model, its concepts are foundational for the development of modern physics and the study of high energy particles.