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Extreme quantum tunneling

Look at the number we have: 10¹⁰⁷⁶. There is a double exponential. This number is huge, but what does it mean?

Quantum tunneling takes exponentially longer with higher and higher potential wells (or farther and farther distances). Roughly speaking, the time taken to cross a barrier of distance $d$ and height $E$ is proportional to $exp(d \frac {\sqrt {E m}} {\hbar \sqrt {2}})$, where $m$ is the mass of a particle. This means that, for a 1eV barrier, each 852 femtometers will approximately halve the probability if the particle is an iron nucleus.

Conversely, the "size" of the barrier in units of $d \frac {\sqrt {E m}} {\hbar \sqrt {2}}$ is roughly the natural log of the time taken. Which is $2.3\times 10^{76}$ (it does not matter if the original number is in years or zepto-seconds). Such a number is still huge even on a macroscopic scale.

This is about enough for $10^{46}$ nuclei (1% the mass of the moon) to simultaneously overcome a barrier $6000m$ wide (the radius of a sphere enclosing about this many nuclei at white-dwarf densities) and the Planck energy of $10^{27} eV$ high, as they all crush together into a small black hole which can then eat the white dwarf.

Doubly exponentially large numbers are contrary to our intuition about what cannot happen. In this case, "quantum mechanical tunneling can't happen at large scales". But we also see similar effects with thermodynamics and monkeys on a typewriter. But our intuition is right: this will never be observed. Even in the unlikely event of protons not decaying, all sources of energy will run dry long before we can see it. Even mechanisms that "passively wait until it happens" are easily defeated by such huge numbers.

Extreme quantum tunneling

Look at the number we have: 10¹⁰⁷⁶. There is a double exponential. This number is huge, but what does it mean?

Quantum tunneling takes exponentially longer with higher and higher potential wells (or farther and farther distances). Roughly speaking, the time taken to cross a barrier of distance $d$ and height $E$ is proportional to $exp(d \frac {\sqrt {E m}} {\hbar \sqrt {2}})$, where $m$ is the mass of a particle. This means that, for a 1eV barrier, each 852 femtometers will approximately halve the probability if the particle is an iron nucleus.

Conversely, the "size" of the barrier in units of $d \frac {\sqrt {E m}} {\hbar \sqrt {2}}$ is roughly the natural log of the time taken. Which is $2.3\times 10^{76}$ (it does not matter if the original number is in years or zepto-seconds). Such a number is still huge even on a macroscopic scale.

This is about enough for $10^{46}$ nuclei (1% the mass of the moon) to simultaneously overcome a barrier $6000m$ wide (the radius of a sphere enclosing about this many nuclei at white-dwarf densities) and the Planck energy of $10^{27} eV$ high, as they all crush together into a small black hole which can then eat the white dwarf.

Doubly exponentially large numbers are contrary to our intuition about what cannot happen. In this case, "quantum mechanical tunneling can't happen at large scales". We also see similar effects with thermodynamics and monkeys on a typewriter. But our intuition is right: this will never be observed. Even in the unlikely event of protons not decaying, all sources of energy will run dry long before we can see it. Even mechanisms that "passively wait until it happens" are easily defeated by such huge numbers.

Extreme quantum tunneling

Look at the number we have: 10¹⁰⁷⁶. There is a double exponential. This number is huge, but what does it mean?

Quantum tunneling takes exponentially longer with higher and higher potential wells (or farther and farther distances). Roughly speaking, the time taken to cross a barrier of distance $d$ and height $E$ is proportional to $exp(d \frac {\sqrt {E m}} {\hbar \sqrt {2}})$. This means that, for a 1eV barrier, each 852 femtometers will approximately halve the probability.

Conversely, the "size" of the barrier in units of $d \frac {\sqrt {E m}} {\hbar \sqrt {2}}$ is roughly the natural log of the time taken. Which is $2.3\times 10^{76}$ (it does not matter if the original number is in years or zepto-seconds). Such a number is still huge even on a macroscopic scale.

This is about enough for $10^{46}$ nuclei (1% the mass of the moon) to simultaneously overcome a barrier $6000m$ wide (the radius of a sphere enclosing about this many nuclei at white-dwarf densities) and the Planck energy of $10^{27} eV$ high, as they all crush together into a small black hole which can then eat the white dwarf.

Doubly exponentially large numbers are contrary to our intuition about what cannot happen. In this case, "quantum mechanical tunneling can't happen at large scales". But we also see similar effects with thermodynamics and monkeys on a typewriter. But our intuition is right: this will never be observed. Even in the unlikely event of protons not decaying, all sources of energy will run dry long before we can see it. Even mechanisms that "passively wait until it happens" are easily defeated by such huge numbers.

Extreme quantum tunneling

Look at the number we have: 10¹⁰⁷⁶. There is a double exponential. This number is huge, but what does it mean?

Quantum tunneling takes exponentially longer with higher and higher potential wells (or farther and farther distances). Roughly speaking, the time taken to cross a barrier of distance $d$ and height $E$ is proportional to $exp(d \frac {\sqrt {E m}} {\hbar \sqrt {2}})$, where $m$ is the mass of a particle. This means that, for a 1eV barrier, each 852 femtometers will approximately halve the probability if the particle is an iron nucleus.

Conversely, the "size" of the barrier in units of $d \frac {\sqrt {E m}} {\hbar \sqrt {2}}$ is roughly the natural log of the time taken. Which is $2.3\times 10^{76}$ (it does not matter if the original number is in years or zepto-seconds). Such a number is still huge even on a macroscopic scale.

This is about enough for $10^{46}$ nuclei (1% the mass of the moon) to simultaneously overcome a barrier $6000m$ wide (the radius of a sphere enclosing about this many nuclei at white-dwarf densities) and the Planck energy of $10^{27} eV$ high, as they all crush together into a small black hole which can then eat the white dwarf.

Doubly exponentially large numbers are contrary to our intuition about what cannot happen. In this case, "quantum mechanical tunneling can't happen at large scales". But we also see similar effects with thermodynamics and monkeys on a typewriter. But our intuition is right: this will never be observed. Even in the unlikely event of protons not decaying, all sources of energy will run dry long before we can see it. Even mechanisms that "passively wait until it happens" are easily defeated by such huge numbers.

Extreme quantum tunneling

Look at the number we have: 10¹⁰⁷⁶. There is a double exponential. This number is huge, but what does it mean?

Quantum tunneling takes exponentially longer with higher and higher potential wells (or farther and farther distances). Roughly speaking, the time taken to cross a barrier of distance $d$ and height $E$ is proportional to $exp(d \frac {\sqrt {E m}} {\hbar \sqrt {2}})$. This means that, for a 1eV barrier, each 852 femtometers will approximately half the probability.

Conversely, the "size" of the barrier in units of $d \frac {\sqrt {E m}} {\hbar \sqrt {2}}$ is roughly the natural log of the time taken. Which is $2.3\times 10^{76}$ (it does not matter if the original number is in years or zepto-seconds). Such a number is still huge even on a macroscopic scale.

This is about enough for $10^{46}$ nuclei (1% the mass of the moon) to simultaneously overcome a barrier $6000m$ wide (the radius of a sphere enclosing about this many nuclei at white-dwarf densities) and the Planck energy of $10^{27} eV$ high, as they all crush together into a small black hole which can then eat the white dwarf.

Doubly exponentially large numbers are contrary to our intuition about what cannot happen. In this case, "quantum mechanical tunneling can't happen at large scales". But we also see similar effects with thermodynamics and monkeys on a typewriter. But our intuition is right: this will never be observed. Even in the unlikely event of protons not decaying, all sources of energy will run dry long before we can see it. Even mechanisms that "passively wait until it happens" are easily defeated by such huge numbers.

Extreme quantum tunneling

Look at the number we have: 10¹⁰⁷⁶. There is a double exponential. This number is huge, but what does it mean?

Quantum tunneling takes exponentially longer with higher and higher potential wells (or farther and farther distances). Roughly speaking, the time taken to cross a barrier of distance $d$ and height $E$ is proportional to $exp(d \frac {\sqrt {E m}} {\hbar \sqrt {2}})$. This means that, for a 1eV barrier, each 852 femtometers will approximately halve the probability.

Conversely, the "size" of the barrier in units of $d \frac {\sqrt {E m}} {\hbar \sqrt {2}}$ is roughly the natural log of the time taken. Which is $2.3\times 10^{76}$ (it does not matter if the original number is in years or zepto-seconds). Such a number is still huge even on a macroscopic scale.

This is about enough for $10^{46}$ nuclei (1% the mass of the moon) to simultaneously overcome a barrier $6000m$ wide (the radius of a sphere enclosing about this many nuclei at white-dwarf densities) and the Planck energy of $10^{27} eV$ high, as they all crush together into a small black hole which can then eat the white dwarf.

Doubly exponentially large numbers are contrary to our intuition about what cannot happen. In this case, "quantum mechanical tunneling can't happen at large scales". But we also see similar effects with thermodynamics and monkeys on a typewriter. But our intuition is right: this will never be observed. Even in the unlikely event of protons not decaying, all sources of energy will run dry long before we can see it. Even mechanisms that "passively wait until it happens" are easily defeated by such huge numbers.