The mass of the electron helps determine the size of atoms and the behaviour of chemistry. Changing it opens up a wide range of possible simulations that explore how matter, materials and even life itself could be transformed.

The electron is a negatively charged elementary subatomic particle that has a mass of approximately 9.109×10⁻³¹ kilograms. The relation between the mass of the electron and the mass of the proton is ≈1/1836. The electron exhibits both particle and wave-like properties.

The hydrogen atom consists of a positively charged proton in the nucleus and a negatively charged electron bound to orbit the nucleus by the Coulomb force that for a system of a proton and an electron states

where Ɛ₀≈8.854×10⁻¹² is the vacuum permittivity, q_e is the charge of the proton and the absolute value of the electron’s charge and r is the distance between these charges. 

A classical approach to the hydrogen atom would imply the following (Tuominen, Kimmo. Kvanttifysiikan Perusteet). A Danish physicist Niels Bohr thought the electron to orbit the proton in a circular orbit of radius r. Assuming that the proton is extremely massive compared to the electron and does not move at all, the relation between the centripetal acceleration a=−v²/r and the Coulomb force that is creating it, is

where m is the electron’s mass. The Coulomb force follows from the potential

Hereby the kinetic and potential energy of the electron can be presented as

Bohr believed that - unlike in a fully classical approach - v and r could get only certain values. He set the angular momentum of the electron L= mvr to only get the values of L=nℏ where n= 1,2,3, .... For the radius of the electron’s orbit we get

Since the angular momentum L is quantized the possible orbits are

where a₀ is the Bohr radius, and defined to be

In reality, electrons do not orbit the nucleus in circular orbits with an unchanging radius. The Bohr radius, the radius when n= 1, gives approximately the most probable distance between the electron and the nucleus when the hydrogen atom is in its ground state. 

The energy states of hydrogen are expressed as

The Bohr radius a₀  gives the most probable distance between the proton and the electron in the ground state of a hydrogen atom. In a way, this could be thought of as the size of the atom. It’s current value is approximately 0.529×10⁻¹⁰ meters.

The Bohr radius is inversely proportional to the mass of the electron. When the mass of the electron increases, a₀ decreases and so the most probable distance between the nucleus and the electron becomes smaller. When the mass of the electron decreases the opposite happens, the Bohr radius grows.

For the Bohr radius to be one meter, the mass of the electron would have to be 4.8×10⁻⁴¹ kilograms.

The energy levels of the hydrogen atom are given above. The ground state energy of an electron, when n= 1, is−13.6eV. This means that it would take 13.6eV to separate the electron and the proton since this situation is defined to have an energy of 0eV. In other words Eₙ→0 eV when n→ ∞.

The energy of the ground state is directly proportional to the mass of the electron. Increasing the mass would result in a situation where more energy would be needed to separate the electron and the proton completely. If the mass of the electron becomes k·m_(initial), where k >0, the new energy states would simply be

The electron can get excited to a higher energy level by absorbing a photon. This absorption can happen only when the energy of the photon equals the energy difference of the old and the new energy level. To excite an electron in the ground state completely out of the atom, one would need a photon with an energy of 13.6eV. This corresponds to electromagnetic radiation with a wavelength of approximately 91 nanometers, which falls in the range of ultraviolet (UV) light. 

The energy of a photon is given by

where c is the speed of light, h is the Planck constant and λ is the wavelength of the electromagnetic radiation. The amount of energy that would be needed to lift an electron out of the atom is

A human being consist mostly of water. For the purpose of understanding the phenomena present, we can think of water, H₂O, to be formed of 100% hydrogen. Therefore, we approximate humans to be fully made out of hydrogen. One gram of hydrogen has approximately 6×10²³ atoms in it. A person of 60 kilograms has 60×10³×6×10²³= 12×10²⁷ hydrogen atoms in them. A light source that emits deep blue/violet light (of a wavelength of 400nm) has photons of an energy of approximately 3eV. For that light to be able to ionise a hydrogen atom and lift the electron away from the atom, all the way from the ground state, one would need that ground state to have an energy of−3eV. Inserting n= 1 and Eₙ=−3eV to equation 28 we get k≈0.2. The mass of the electron would have to become 0.2 times its initial value for photons of 3eV to be able to ionise hydrogen in its ground state. To ionise a whole human with these 3eV photons, one would need one photon for each atom, so 12×10²⁷ photons.

Let’s say the light source is a 60-watt light bulb that only emits 400nm light. In other words, every second 60 joules of energy is emitted. This corresponds to ≈3.75×10²⁰ eV. One photon carries 3eV, so every second 1.25×10²⁰ photons are emitted. It would take 9.6×10⁷ seconds to ionise all the atoms in a human with this light source. This corresponds to over three years assuming that each emitted photon ionises an atom.

The relation between the mass of the electron and the mass of the proton is approximately 1/1836. As a result, small changes in the mass of the electron will not make a noticeable contribution to the total mass of an object. 

If the mass of the electron increased to 1836 times its current one, it would have the same mass as the proton. In this case the mass of the objects that have the same amount of protons and electrons would seem to double, since now basically only the protons have an impact.

There is a non-zero probability of finding the electron inside the nucleus of the hydrogen atom. The nucleus of the hydrogen atom consists of a single proton that has a charge radius of approximately r₀ ≈ 10⁻¹⁵ meters. For the ground state the electron’s wave function is

where a₀ is the Bohr radius and r is the distance from the origin (centre of the proton). If we approximate the electron’s wave function to be constant over the nucleus, the probability of finding the electron inside the proton’s volume is

where we have integrated over the proton’s volume. The electron’s wave function in the proximity of the nucleus that rests in the origin is

Hereby, the probability of finding the electron inside the proton’s volume is

Inserting the Bohr radius a₀ (as defined above) we get

The probability of finding the electron inside the nucleus is proportional to the cube of the electron’s mass.

In a situation where the electron would be found inside the charge radius with close to a 100% certainty we would have

For the hydrogen atom the charge radius r₀ is approximately 10⁻¹⁵. Solving mₑ we find it to be approximately 48000 times the initial electron mass. Thus, the mass of the electron would have to become almost fifty thousand times its initial mass so that the electron would be found inside the charge radius with close to a 100 percent certainty. The Bohr radius in this case would be approximately the same as the hydrogen’s charge radius.

The Bohr radius for heavier elements at the lowest energy level is defined as

where mₑ is the mass of the electron, qₑ  is the elementary charge and Z is the atomic number of the element, which equals the number of protons in the nucleus.

For heavier elements (that have an equal number of protons and neutrons) a good approximation of the charge radius is

where γ is an empirical constant with the approximate value of 10⁻¹⁵. For the probability of finding the electron inside the charge radius to be close to one, and the mass of the electron to remain at its initial value, the atomic number Z would have to be 2731. In this case the Bohr radius would be a₀ ≈ 1.9×10⁻¹⁴ and the charge radius would be r₀ ≈ 1.76×10⁻¹⁴.

If the Bohr radius and the charge radius had the same value, and the mass of the electron remained at its initial value, the atomic number Z would have to be 2933. The Bohr radius and the charge radius would have the value of a₀ = r₀ ≈ 1.8×10⁻¹⁴. However, in this case, the probability of finding the electron inside the charge radius seems to be over one. 

When the atomic number grows, the charge of the nucleus increases as the atomic number equals the amount of protons in the atom. Hereby, the size of the nucleus grows as well. The Bohr radius decreases as Z becomes larger as the increased charge in the nucleus is pulling the electrons closer.

According to wave-particle duality, particles can also be represented as waves. The wavelength of these matter waves is the de Broglie wavelength that is given by

where h is the Planck constant and p is the momentum of the electron

When the velocity of the electron is small, v²/c²→0 and the momentum is simply

The de Broglie wavelength is inversely proportional to the mass of the electron. By increasing the mass of the electron its wavelength decreases. The wavelike properties of matter become evident at the scale of the de Broglie wavelength in question. The larger the wavelength the easier it is to make these phenomena visible. So when the mass of the electron is smaller, its wavelength is larger and for example observing an interference pattern created by the electron going through two slits can be done on a larger scale.

For an electron moving with a velocity of 1 m/s, its mass would have to be approximately 7×10⁻⁴ times the initial electron mass for the deBroglie wavelength to have a value of one meter.

The ratio between the mass of the electron and the mass of the proton is approximately 1/1836. We look at what would happen if the mass of the electron became the same as the mass of the proton.

The Bohr radius of the hydrogen atom is inversely proportional to the mass of the electron. As the electron’s mass increases the most probable distance between the electron and the nucleus decreases. In a situation where the proton and the electron had the same mass, the Bohr radius would be

The Bohr radius would be further away from the centre of the nucleus than the charge radius (that can be thought of as the size of the nucleus/proton), meaning that the electron’s most probable distance from the nucleus would be ’outside’ the proton.

The probability that the electron is found inside the charge radius r₀ is