Waves are something that have *no mass* and move at the *maximum speed*, mass **m** = 0. speed **c** = 1. So whats their *momentum*? p = m.v = 0? **Right**?

**No**. For *pure* waves; *momentum does not come from mass*. It comes **only** from *motion*.

(pure wave; they do **not** have mass)

For matter waves, on the other hand, momentum comes in **two** ways, *mass* as well as *motion*.

(impure, now they **have** *mass*)

Albert Einstein *recognized* this fact and derived his relation;

This relation is called as Einstein’s relativistic equation, also Einstein’s mass-energy relation. But more appropriately mass-energy-momentum relation.

Let us consider **E** as the *hypotenuse*, **p** and **m**; as *base* or *perpendicular* as is your choice.

Then is Pythagoras Theorem; **when p** is *momentum *and **m** is *mass*.

For pure waves such as **photon** … *the quanta of light*, **m** = **0**.

Hence the Pythagorean Triangle is now one**,** where the *mass* side is arbitrary **small**. Thus **E** = **p**.

(A triangle with **two** sides because one side is as small as we wish)

This **E** = **p** relation is called as **on-shell** *condition*. A reason why nothing moves faster than waves. People have a habit of saying *nothing is faster than light* for this reason, as; light is also wave.

So in general, *Energy is equivalent to mass ( m) and momentum (p)*. And not just

*mass*, as

**is**popularly held.

What happens when you say; *I divide Energy into two parts E1 and E2*, such that

**E1**

*corresponds*to

**m**and

**E2**to

**p**.

Then **E** = **E1** = **m** comes only when there are particles that are **not** *moving*, so **p** = **0**, note that this is **never** the case with what I called as waves or *pure waves*, such as *photons*. Photons will always have non zero **p **and zero** m**. As a consequence **E1** = **0**, **E** = **E2** = **p**.

In the **E** = **E1** = **m** case we say *energy is equivalent to mass*. Since, **E** = **m**, this is called as; *Einstein’s mass-energy equivalence relation*. Also written as

**E**=

**m**.

**c**-square.

(Also **E** = **p** can be written as **E** = **p**.**c**)

People always say this and never care to mention in general; * energy is equivalent to momentum and mass*, that is energy comes from

*mass content*as well as

*motion*.

For pure waves, thus, *energy is equivalent to momentum* **only**. As we saw above **E** = **p**.

Thus **m** = **0** does **not** make *momentum* **zero**. Momentum **equals** to *energy* and comes from *waves*. Waves are constantly moving, at speed 1, that is; **3**.**00** x **10** ^ **8** m/s.

All **particles**, in the Universe, thus; lie in-between **m** = **0** and **v** = **1**.

So *photons* are more wave like (in Q-Mechanics) because they have **zero** *mass*. But *electrons* which have a **small** mass are more particle like, than waves, even if both photons and electrons are supposed to be wave-particles in terms of Q-Mechanics.

That is why *the more mass one has the more particle like one will behave*, even though inherently one has wave properties as per Q Mechanics.

Although **de**–**Broglie** connects *momentum* to *wavelength* of **particles**, for *pure* waves such as *photons* or very small mass particles, the *momentum coming from mass* is very **small** or **zero**. So they show wave properties based on their **energy** = **momentum**, and **not** *momentum coming from mas*s*.*

But for more and more **massive** *particles* the motion is at very **low** *speed*, compared to maximum **1** = **300**,**000** km/s, the *momentum *would be* coming from mass*.

A stone, say of mass 20 gm, thrown at **20** m/s, hardly shows any wave behavior. Its *wave momentum* is conspicuously lacking, due to **heavy** *mass*. But an electron, moving at **10**% of *speed of light* has appreciable *wave behavior* because its mass is quite quite **small** and its closer to actual *pure waves like photon*. It has significant *wave momentum* even though *momentum due to mass* could be almost *negligible*.

That last remark is a bit *tricky* because one **has to** do actual calculation, but I am highlighting the idea, for the above discussion.

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