# E=hf interpretation

I know what the letters mean, E = Energy of the photon, h = planck's constant, f = frequency of the photon.
But what does it mean for a particle to have a frequency, something that I'd associate with a wave. And what can you think planck's constant is representing?
Any replies would be much appreciated, thanks.

## Answers and Replies

Nugatory
Mentor
You might expect that when a light wave encounters matter, it delivers its energy more or less evenly across the entire illuminated surface. It doesn't. Instead, the energy is delivered in discrete lumps to single points, and we say that each of these represents a single photon arriving. The ##E=hf## relationship tells us how much energy will be in each discrete lump, given the wavelength and frequency of the incoming light.

If the light is reasonably bright, we never notice this lumpy behavior. There are so many photons that we end up with a uniform even distribution, just as a pouring rainstorm will uniformly wet everything even though the water is delivered in individual drops. (Warning - I have just pushed this raindrops analogy as far it goes. Light is not a stream of photons the way a rainstorm is a stream of raindrops - it just happens to interact with matter sort of that way).

atyy
In quantum mechanics, we (roughly speaking) associate a wave to each particle. The frequency of the particle is the frequency of the associated wave. An intuitive (but not strictly correct) picture for a single particle is that the particle is really not a particle, but a wave that is tightly peaked around a single location. Because the wave is tightly peaked around a single position, we can think that there is a approximately well localized particle at the position of the peak of the wave. The tightly peaked wave can be thought of via Fourier decomposition as being made of many sinusoidal waves that are not well localized and have a frequency.

For later reference: This picture becomes less correct as one goes to many particles, but even then we can associate a wave with each particle. You don't have to understand that at the moment, but for reference this is the idea that the state of the whole system is a tensor product of the single particle states, and the wave can be taken in the position basis. Even in quantum field theory, we can take the wave-particle duality to be the particles of the Fock space, and the wave as the equation of motion of the field observable.

stevendaryl
Staff Emeritus
I know what the letters mean, E = Energy of the photon, h = planck's constant, f = frequency of the photon.
But what does it mean for a particle to have a frequency, something that I'd associate with a wave. And what can you think planck's constant is representing?
Any replies would be much appreciated, thanks.

Operationally, if you have a source of monochromatic light--blue light or whatever--then that light will have a certain characteristic frequency. When you measure the energy of photons coming from such a source, they will be found to have the corresponding energy.

You might expect that when a light wave encounters matter, it delivers its energy more or less evenly across the entire illuminated surface. It doesn't. Instead, the energy is delivered in discrete lumps to single points, and we say that each of these represents a single photon arriving. The ##E=hf## relationship tells us how much energy will be in each discrete lump, given the wavelength and frequency of the incoming light.

If the light is reasonably bright, we never notice this lumpy behavior. There are so many photons that we end up with a uniform even distribution, just as a pouring rainstorm will uniformly wet everything even though the water is delivered in individual drops. (Warning - I have just pushed this raindrops analogy as far it goes. Light is not a stream of photons the way a rainstorm is a stream of raindrops - it just happens to interact with matter sort of that way).
Thankyou very much.
However I have to say that I'm still confused about the fact that in order for the E=hf equation to work you have to assume light is a stream of particles however you also assume light is a wave for you to measure the frequency. And what do you mean by light isn't a stream of particles in the sense that raindrops are a stream of particles?
Thanks.

jtbell
Mentor
Light is neither a (classical) wave nor a (classical) particle. We describe it as a quantum field, with some aspects that are reminiscent of particles and some aspects that are reminiscent of waves.

In quantum mechanics, we (roughly speaking) associate a wave to each particle. The frequency of the particle is the frequency of the associated wave. An intuitive (but not strictly correct) picture for a single particle is that the particle is really not a particle, but a wave that is tightly peaked around a single location. Because the wave is tightly peaked around a single position, we can think that there is a approximately well localized particle at the position of the peak of the wave. The tightly peaked wave can be thought of via Fourier decomposition as being made of many sinusoidal waves that are not well localized and have a frequency.

For later reference: This picture becomes less correct as one goes to many particles, but even then we can associate a wave with each particle. You don't have to understand that at the moment, but for reference this is the idea that the state of the whole system is a tensor product of the single particle states, and the wave can be taken in the position basis. Even in quantum field theory, we can take the wave-particle duality to be the particles of the Fock space, and the wave as the equation of motion of the field observable.
That's very interesting, thanks. Although I have to admit I don't totally follow you. I haven't really learned quantum physics in depth yet as I'm only at A-Level level (high school). But it's clarifying in a really paradoxical way because I thought there was something really obvious I'm missing but now I see theres a lot more to it than what I've been learning about which is probably really superficial.
If you wouldn't mind just one more question: is planck's constant just something that makes the equation work or is it something deeper than that?

Nugatory
Mentor
Thankyou very much.
However I have to say that I'm still confused about the fact that in order for the E=hf equation to work you have to assume light is a stream of particles however you also assume light is a wave for you to measure the frequency.
You don't have to assume that it is a stream of particles. Think about it as a wave that delivers its energy to its target in an odd and surprising way.

And what do you mean by light isn't a stream of particles in the sense that raindrops are a stream of particles?
Thanks.
I mean exactly that. When people first hear that photons are "particles" of light, the mental image that forms is a little tiny bullet (if it's moving fast) or grain of sand (not moving so fast) or other small object travelling through space. That picture is hopelessly misleading, as a photon has neither a position nor a path except at the moment that it interacts with something else.

If you can get hold of Richard Feynman's book "QED: The strange theory of light and matter", I highly recommend it.... No advanced math required.

Nugatory
Mentor
is planck's constant just something that makes the equation work or is it something deeper than that?

It's deeper than that, in the sense that it appears throughout quantum mechanics, in many more places than just the frequency/energy relationship for light.

It's also "just something that makes the equations work", in the sense that its value is determined by experiment. However, that's true of a lot of important constants. For example, you are probably familiar with Newton's gravitation law ##F=Gm_1m_2/r^2##; but did you notice that ##G## is "just" the number that makes this equation work?

You don't have to assume that it is a stream of particles. Think about it as a wave that delivers its energy to its target in an odd and surprising way.

I mean exactly that. When people first hear that photons are "particles" of light, the mental image that forms is a little tiny bullet (if it's moving fast) or grain of sand (not moving so fast) or other small object travelling through space. That picture is hopelessly misleading, as a photon has neither a position nor a path except at the moment that it interacts with something else.

If you can get hold of Richard Feynman's book "QED: The strange theory of light and matter", I highly recommend it.... No advanced math required.
Wow I feel brainwashed. I've always thought of the photons as particles in the sense of a bullet. I've even seen documentaries talking about photons with animations of yellow spheres gliding through the air. To be honest I'm doubtful of the truth of most of the things I'm learning in my A-Level quantum physics course because it all seems so oversimplified and shallow. I think I should wait for Physics at University for the accurate picture. Or in the meantime read QED :)

atyy
That's very interesting, thanks. Although I have to admit I don't totally follow you. I haven't really learned quantum physics in depth yet as I'm only at A-Level level (high school). But it's clarifying in a really paradoxical way because I thought there was something really obvious I'm missing but now I see theres a lot more to it than what I've been learning about which is probably really superficial.

Yes, what you are probbaly learning is "old quantum theory", which includes Planck's introduction of his constant for blackbody radiation, and Einstein's proposal of E=hf. These were not incorporated into a single coherent framework of quantum mechanics, which stands till this day, until 1927-1928 by Heisenberg, Schroedinger, Born and Dirac. In the new framework, there is no strict concept of "wave-particle duality", but one understands that it is still a good heuristic, even if not a strictly correct building block.

If you wouldn't mind just one more question: is planck's constant just something that makes the equation work or is it something deeper than that?

Planck's constant is both something that just makes his equation work, but because it is a new universal constant, it is also deep. One way to see that is that it is needed to get quantities like the "Bohr radius" (again problematic formally, but very important in setting heuristically what we think the size of an atom is), and the "Planck length"
http://www.math.ucr.edu/home/baez/lengths.html
http://math.ucr.edu/home/baez/planck/node2.html

Nugatory
Mentor
Wow I feel brainwashed.
You are in good company here :)

The problem is that quantum mechanics in general and Quantum Field Theory in particular don't map well onto our common sense (meaning "limited to experience with macroscopic objects") intuitions of how things behave. This makes it very difficult to explain without using the math, and there's enough math involved to make the topic basically inaccessible until a year or so into college.

Many intro-level non-mathematically oriented discussions of quantum mechanics (pretty much anything that talks about "wave particle duality", or Schrodinger's dead and alive cat, or particles going through two slits at the same time) are just sharing the puzzlement that physicists felt when they first started encountering quantum phenomena early in the 20th century.

I think I should wait for Physics at University for the accurate picture. Or in the meantime read QED :)
Do both.

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Thanks for your replies everybody, much appreciated. This has given me a lot more insight into what I'm learning about and the inspiration for learning physics to a higher level, knowing that there's something a lot deeper and more interesting at work. Thanks!