Question 1. Meaning of symbols
I'd have liked a little bit more explanation on what the equation means, especially what the symbols describe. You tell us about the wave function symbol, but not what the other two on the left and the remaining one on the right mean. Carl
Hi Carl.
It’s a great question because although the wavefunction is the strangest and most important part of the equation from the perspective of trying to understand the differences between quantum physics and classical physics, each piece of the equation is enigmatic in its own way and tells us something about characteristics of the microscopic world.
“i”
This is the imaginary number – ‘the square root of -1’, from which we see that quantum theory will be fundamentally built around complex numbers, something that sets it apart immediately from similar types of equations in classical physics. In classical physics we sometimes use complex numbers for convenience, but here they are a core, inextricable, part of the theory.
“ħ”
This is just a number - a constant of nature (Plancks constant is a very small number = 6.626068 × 10-34 m2 kg / s) that determines the microscopic scale at which quantum effects become important. It’s a pity it isn't larger so that we could see these effects with our own eyes.
Psi with a dot on it.
Psi is the quantum state or quantum wavefunction, the source of many of my headaches. From it we can predict the probabilities of outcomes of experiments. It’s a complicated mathematical object that can be very large (in the sense of requiring terabytes of data to store on a computer) even when it’s describing only a very small number of atoms or photons or similar. Although we call it a wavefunction and we often say quantum systems act like waves, the “waviness” does not refer to oscillations in the space around you that you perceive, but rather in some much larger mathematical space. The dot on top indicates that this is the derivative of Psi with respect to time (how Psi is changing in time).
H with a hat on it.
H is another complicated mathematical object (called the Hamiltonian operator) that captures the quantum theory version of energy. Energy comes in many forms, and the Hamiltonian is a quantum mechanical version of the sum of all those energies. The hat denotes that you cannot think of it as a single real number like “total kilowatt hours” that appears on your utility bill - it is a much more structured object, as you might guess by the fact that it needs to “act on” (multiply) the wavefunction Psi to the right of it.
As for what the equation “means”, it is basically the equation which governs how microscopic things such as atoms move around, and it dictates what we can predict about where we will find them after they have moved around for some time. To see why energy would come into it, think about a stationary ball at the top of a hill: at that point we would say is has gravitational potential energy; if it now rolls down the hill it converts that energy into what we call kinetic energy – the energy of its motion. In the quantum world motions are still governed by inter-converting energy, it’s just that they are a heck of a lot more complicated.
Terry
I'd have liked a little bit more explanation on what the equation means, especially what the symbols describe. You tell us about the wave function symbol, but not what the other two on the left and the remaining one on the right mean. Carl
Hi Carl.
It’s a great question because although the wavefunction is the strangest and most important part of the equation from the perspective of trying to understand the differences between quantum physics and classical physics, each piece of the equation is enigmatic in its own way and tells us something about characteristics of the microscopic world.
“i”
This is the imaginary number – ‘the square root of -1’, from which we see that quantum theory will be fundamentally built around complex numbers, something that sets it apart immediately from similar types of equations in classical physics. In classical physics we sometimes use complex numbers for convenience, but here they are a core, inextricable, part of the theory.
“ħ”
This is just a number - a constant of nature (Plancks constant is a very small number = 6.626068 × 10-34 m2 kg / s) that determines the microscopic scale at which quantum effects become important. It’s a pity it isn't larger so that we could see these effects with our own eyes.
Psi with a dot on it.
Psi is the quantum state or quantum wavefunction, the source of many of my headaches. From it we can predict the probabilities of outcomes of experiments. It’s a complicated mathematical object that can be very large (in the sense of requiring terabytes of data to store on a computer) even when it’s describing only a very small number of atoms or photons or similar. Although we call it a wavefunction and we often say quantum systems act like waves, the “waviness” does not refer to oscillations in the space around you that you perceive, but rather in some much larger mathematical space. The dot on top indicates that this is the derivative of Psi with respect to time (how Psi is changing in time).
H with a hat on it.
H is another complicated mathematical object (called the Hamiltonian operator) that captures the quantum theory version of energy. Energy comes in many forms, and the Hamiltonian is a quantum mechanical version of the sum of all those energies. The hat denotes that you cannot think of it as a single real number like “total kilowatt hours” that appears on your utility bill - it is a much more structured object, as you might guess by the fact that it needs to “act on” (multiply) the wavefunction Psi to the right of it.
As for what the equation “means”, it is basically the equation which governs how microscopic things such as atoms move around, and it dictates what we can predict about where we will find them after they have moved around for some time. To see why energy would come into it, think about a stationary ball at the top of a hill: at that point we would say is has gravitational potential energy; if it now rolls down the hill it converts that energy into what we call kinetic energy – the energy of its motion. In the quantum world motions are still governed by inter-converting energy, it’s just that they are a heck of a lot more complicated.
Terry