## An introduction to momentum and wave functions

Before the turn of the 20th century, the greatest minds of the classical world were met with limitations. Newton, Galileo, Euler, Bernoulli, and Lagrange, were all confined to classical postulates. The world, as we knew it, was yet to discover the atomic dimension through the quantum lens.

A century later, in 1925, modern quantum mechanics was born. Schrodinger, Dirac, Einstein, Bohr, Feynman, and Planck— among others — laid the fundamentals of quantum theory. To a great extent, they brought forward a brilliantly accurate model. And like any remarkable theory, quantum physics challenged the status quo. The world saw an unprecedented influx of paradigm shifts. The greatest minds of the 20th century went head-to-head, back and forth, debating over what made sense and what didn’t.

On the smallest scale, the world is weird. When things get down to the size of an atom, the physics that governs us no longer applies. We throw classical mechanics out the window and quantum physics takes over. And since it’s quantum physics, new rules are needed, and to describe them, new counterintuitive equations. The idea of an objective reality seems foolish; replaced with notions like probability distributions rather than deterministic outcomes, wavefunctions rather than positions and momenta, Heisenberg uncertainty relations rather than individual properties, complex rather than real numbers. The list goes on but what I’d like for you to take away from this is that quantum mechanics is not an improvement on classical mechanics. It’s a completely new idea that replaces our classical understanding.

Around when Schrodinger was messing with wave functions, it was well known that quantum mechanics, to be successful, required complex systems. The imaginary number crept its way into equations and showed up virtually everywhere. Nevertheless, it worked. But there came a question:

“How does something complex describe something real?”

One such *complexity *was momentum. In classical physics, we calculate an object’s momentum by multiplying its mass by its velocity. But how do we deal with momentum in quantum mechanics? Is it again, just mass and velocity? Or is something else altogether?

The first difference I think you would’ve already taken note of is that in quantum theory, we deal with probabilistic positions — not definite ones. We can, with **certainty**, only describe the **probability** of a particle being found at a particular point in space. Therefore, a particle’s position, through space and time, is defined using a wave function.

The wave function of a system is a mathematical function that contains all the information we can know about a system.

It’s important to note that the wave function, when squared, can be used to calculate the probability of finding a particle at different positions in space. If you want to find out more about wave functions, check this out:

Well, as we already established, our wave function, Ψ, tells us how likely we are to find a particular particle in different regions of space in our system. Let’s say that our **quantum** system is a single electron. In our system, like the one we have below, a particle can move forwards or backward since it’s confined to one dimension.

Ψ² ∝ probability distribution of finding a particle at a particular point.

It’s this function like we established, that is the important one. Specifically, this function is telling us — Ψ² is telling us — that at r=2 because Ψ² is large, we’re most likely to find our electron. Whereas at 4, Ψ² is small and so we’re least likely to find our electron. Now there are a few subtleties to this but for our purpose, all we care about is that the Ψ² function is telling us the probability distribution of finding our electron in our one–dimensional system. All that matters is that if we wanted to, we could find the Ψ of our system. It’s only when we make a measurement on the particle, that we cause a collapse in the wave function, and thus know a certain value for the particle’s position. This value need not necessarily be the position that has the greatest probability. It could simply be 3, or 5, maybe even 9.

So with the quantum mechanical difference that we now use wave functions instead of definite positions, can we still say that momentum is a product of mass and velocity? Well, no.

Unsurprisingly, the wave function is also directly linked with momentum. With a simple mathematical transformation (usually not that simple), we can calculate the probability of finding our particle with different values of momentum in space.

So a question we can ask is, how do we mathematically deal with the idea of probabilistic momentum? In classical physics, we don’t need probabilities. We can describe momentum with certainty — the product of mass and velocity. But in quantum physics, the maths primarily deals with wavefunctions. Since the system is described by a function, we can apply a “measurement operator” to that function. This is the real-life equivalent of measuring the particle. So, this operator, when applied to a wave function, yields the value of what we’d find in an experiment.

Here, P represents the momentum operator and lambda (the weird thing on the right) is the eigenvalue. This equation here is unimaginatively known as an eigenvalue equation. When we apply a momentum operator to a wave function, the eigenvalue (which is one of the things on the right-hand side) is the value we would experimentally witness. Now, eigenvalues are a little more elaborate than just this, but for this article, all we really need to know is that we can apply a momentum operator to a wave function if we wish to.

But what does the momentum operator, that we use in the eigenvalue equation, actually look like? Is it mass and velocity? Well, evidently not.

With, i, the imaginary number, **ℏ**, the reduced Planck constant, and a partial derivative with respect to x as part of the expression. The momentum operator is a little daunting but looks exactly like that.

Why is there a minus sign along with an imaginary number? Well, all of this comes from the fact that, mathematically, we apply this operator to the particle’s wave function rather than the particle itself. One of the simplest waves we can consider in quantum mechanics has a mathematical form that looks something like this:

Well, the first thing you can observe is that e, the transcendental number, appears in the exponential function, e to the x:

Now, by no stretch of the imagination, can we consider this to be a “wave”. But the quantity, i, changes all of that. We credit this to Euler’s form:

Needless to say, both the cosine and the sine functions are waves.

Sinusoidal functions — sine and cosine — can be written in Euler’s form with the imaginary number in the exponent.

All of this is just to say that although exponential functions do not look like a wave, e to the power i[something] does look like a wave.

Now, in the wave function we’re considering, p is the momentum of the particle, x is, of course, the position, E is the energy, and t is time. Putting all of this together, we got:

Again, all of this refers to a particular type of wave. We’ve omitted other directions such as y, and z. With that, we’ve also conveniently excluded more complex forms of a wave. The function shown above takes the form of a sinusoidal curve. Of course, that’s the simplest. We could also, for example, consider a wave function that looked like this:

This would be a nightmare to work with. So instead, we can break this “more complicated” function down into smaller parts and write it as a sum of simpler wave functions.

Now, for those of you familiar with differentiation, we can find the (partial) derivative of Ψ with respect to *x*** . **But it’s completely alright if you’re lost here since the essence of the idea remains the same.

To do this, we assume that **nothing else, **apart from *x, *depends on *x*. In essence, what we’re saying is that momentum, energy, and time, are all independent of the position. For the uninitiated, one thing that you’d need to know is that differentiation is a mathematical operation that allows us to find the gradient of a function at each point. In other words, what we’re really doing, is calculating the rate at which the wave function changes with respect to *x. *We can take our wave, differentiate it, and the result would tell us the gradient at each point. This may seem unimportant and fairly trivial but that’s because it is. We don’t really care about the gradient of the wave function. It tells us virtually nothing. The only reason we even differentiate it is because we find this:

The interesting thing to note is that our differentiated wave function contains the original wave function.

This means that we can rewrite this equation to solve for the momentum.

This is where the mathematical formulation of momentum comes from. What we’re seeing here is that momentum is a little more complicated in the quantum world. It’s worth noting though that the logic behind momentum is still the same — the conservation of momentum still applies. With that, I hope you understood where quantum momentum comes from

## FAQs

### What is the momentum in quantum mechanics? ›

In quantum mechanics, momentum is defined as **a self-adjoint operator on the wave function**. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables.

**What is the hardest question in quantum physics? ›**

Quantum Gravity

The biggest unsolved problem in fundamental physics is **how gravity and the quantum will be made to coexist within the same theory**. Quantum Gravity [1] is required to make the whole of physics logically consistent.

**How is momentum measured in quantum mechanics? ›**

This is the essence of measurement in quantum mechanics. Classical momentum can be obtained simply by **measuring the time an object takes to pass between two stationary detectors ('time-of-flight'), finding the velocity and multiplying by the mass**.

**Why quantum mechanics is so difficult? ›**

Quantum mechanics is deemed the hardest part of physics. Systems with quantum behavior don't follow the rules that we are used to, they are hard to see and hard to “feel”, can have controversial features, exist in several different states at the same time - and even change depending on whether they are observed or not.

**What is momentum easy explanation? ›**

Momentum is defined as **the quantity of motion of the body**.

It is measured by “mass × velocity”, as momentum depends upon velocity, and it depends on the direction of the motion of the body as well. Momentum is a vector quantity since velocity is a vector while mass is scalar.

**What is momentum simple answer? ›**

Momentum is **the quantity that is used to describe the state of motion of an object with a non-zero mass**. Hence, momentum is applicable to any moving object. If m is the mass of an object and v → is the velocity with which this body travels, then momentum can be expressed as p → = m v → .

**What is the holy grail of physics? ›**

“Finding a deviation from general relativity is the holy grail in physics,” says Dr Dominic Dirkx of the section Astrodynamics and Space Missions.

**What is the biggest mystery in physics? ›**

**plications” with a brief explanation/justification.**

- 1 Quantum Gravity. The biggest unsolved problem in fundamental physics is how gravity and the. ...
- 2 Particle Masses. ...
- 3 The “Measurement” Problem. ...
- 4 Turbulence. ...
- 5 Dark Energy. ...
- 6 Dark Matter. ...
- 7 Complexity. ...
- 8 The Matter-Antimatter Asymmetry.

**Why did Einstein refuse quantum mechanics? ›**

Einstein always believed that everything is certain, and we can calculate everything. That's why he rejected quantum mechanics, **due to its factor of uncertainty**.

**What is the rule of momentum? ›**

**Momentum is equal to the mass of an object multiplied by its velocity** and is equivalent to the force required to bring the object to a stop in a unit length of time. For any array of several objects, the total momentum is the sum of the individual momenta.

### Why momentum in quantum mechanics is imaginary? ›

Definition (position space)

where ∇ is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit. will change its value. Therefore, **the canonical momentum is not gauge invariant, and hence not a measurable physical quantity**.

**How is momentum determined? ›**

Momentum depends upon the variables mass and velocity. In terms of an equation, **the momentum of an object is equal to the mass of the object times the velocity of the object**. where m is the mass and v is the velocity.

**What is the hardest branch of physics? ›**

Atomic Physics is considered one of the hardest branches of Physics.

**What is the hardest physics equation? ›**

Yet only one set of equations is considered so mathematically challenging that it's been chosen as one of seven “Millennium Prize Problems” endowed by the Clay Mathematics Institute with a $1 million reward: the **Navier-Stokes equations**, which describe how fluids flow.

**What is the easiest way to learn quantum mechanics? ›**

**How to understand quantum mechanicsEdit**

- Complex numbers.
- Partial and Ordinary differential equations.
- Integral calculus I-III.
- linear algebra.
- fourier analysis.
- probability theory.

**How do you explain momentum to a child? ›**

One very simple way to demonstrate momentum is to **roll a small ball or toy car down a ramp, so it collides with another ball or toy car at the bottom**. As the ball or car rolls down the ramp, its momentum increases as it picks up speed. The object at the bottom is stationary until the first object collides with it.

**What is momentum explain with examples? ›**

Examples of momentum

**Whenever you toss a ball at someone as well as it smacks him square in the face**. It indicates how difficult it would have been to stop the thing. A baseball is swooping through the air. A large truck is moving. A bullet discharged from such a firearm.

**What does momentum depend on? ›**

Thus, momentum is dependent on the **mass and velocity of the body**.

**How do you write momentum answer? ›**

**Momentum (P)** is equal to mass (M) times velocity (v).

**Did Jesus use the Holy Grail? ›**

The Holy Chalice, also known as the Holy Grail, is in Christian tradition the vessel that **Jesus used at the Last Supper to serve wine**. The Synoptic Gospels refer to Jesus sharing a cup of wine with the Apostles, saying it was the covenant in his blood.

### Who is the godfather of physics? ›

**Newton, Galileo and Einstein** have all been called "Fathers of Modern Physics." Newton was called this because of his famous law of motion and gravitation, Galileo for his role in the scientific revolution and his contributions on observational astronomy, and Einstein for his groundbreaking theory of relativity. Q.

**Does the Holy Grail still exist? ›**

The Holy Grail – the sacred cup Jesus drank from at the Last Supper – is one of the most well-known symbols in Christianity. It's also one of the religion's greatest sources of myth and mystery. Yet despite the Grail's fame, no one is entirely sure where it is or whether it ever existed.

**What is the deepest secret of the universe? ›**

What is **dark energy**? It's one of the universe's biggest mysteries: more remains unknown than known about dark energy. It affects the universe's expansion, so physicists are able to infer that dark energy makes up roughly 68% of the universe and it appears to be somehow tied to the vacuum of space.

**What is the number one rule of physics? ›**

**Newton's first law** states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force.

**What are the paradoxes of quantum physics? ›**

In the history of quantum physics three types of paradox exist: **those that challenge classical physics; those that challenge intuition and common sense; and those that challenge quantum physics itself**.

**Did Nikola Tesla believe in quantum mechanics? ›**

Nikola Tesla, the maverick scientist who **disagreed with the basic assumptions of quantum theory**, developed an aether-based theory of the cosmos and came remarkably close to the Chinese view of nature. Unlike European science, Chinese natural philosophy is based on associative rather than analytical principles.

**Does quantum mechanics violate relativity? ›**

Quantum mechanics embraces action at a distance with a property called entanglement, in which two particles behave synchronously with no intermediary. This nonlocal effect is not merely counterintuitive: **it presents a serious problem to Einstein's special theory of relativity**, thus shaking the foundations of physics.

**Why can't quantum mechanics explain gravity? ›**

**The math is too complicated**. There are simply too many possible configurations of both the interactions and the underlying space-time. We can't make the math simple enough to solve; our mathematical models lose their predictive power.

**What are the three laws of momentum? ›**

In the first law, an object will not change its motion unless a force acts on it. In the second law, the force on an object is equal to its mass times its acceleration. In the third law, when two objects interact, they apply forces to each other of equal magnitude and opposite direction.

**What is the first law of momentum? ›**

Conservation of momentum is a fundamental law of physics which states that the momentum of a system is constant if there are no external forces acting on the system. It is embodied in Newton's first law (the law of inertia).

### Is momentum a force or energy? ›

Kinetic energy is a measure of an object's energy from motion, and is a scalar. Sometimes people think momentum is the same as force. Forces cause a change in momentum, but **momentum does not cause a force**. The bigger the change in momentum, the more force you need to apply to get that change in momentum.

**Is it true that only moving objects have momentum? ›**

ANSWER: FALSE - **An object has momentum if it is moving**. Having mass gives an object inertia. When that inertia is in motion, the object has momentum. If an object does not have momentum, then it definitely does not have mechanical energy either.

**Why does momentum stay constant? ›**

The conservation of momentum states that, within some problem domain, the amount of momentum remains constant; **momentum is neither created nor destroyed, but only changed through the action of forces as described by Newton's laws of motion**.

**Why can photons have momentum? ›**

In addition to being a particle, **light is also a wave**. This allows it to carry momentum, and therefore energy, without having mass.

**How does momentum become force? ›**

Force is a measure of the change of momentum over time. It can be written as F = mass x change in velocity / time. In practical terms, **the momentum of an object increases when a force is acting upon it, because the force is causing it to accelerate, and to have an increase in velocity**.

**Why is momentum imaginary in quantum mechanics? ›**

**Since momentum is a hermitian operator, it must have real eigenvalues**. Instead the exponential is interpreted as a reduction in probability of finding the particle deeper in the well. A components of the wave vector may well be imaginary. Since →p=ℏ→k this means that the eigenvalue of that component of →p is imaginary.

**What is momentum vs inertia? ›**

**Momentum is a vector quantity as it is the tendency of a body to remain in motion.** Inertia is a scalar quantity as it is the resistance offered by the body to any change in its velocity.

**What is the momentum of an electron? ›**

The angular momentum of an electron by Bohr is given by **mvr or nh/2π** (where v is the velocity, n is the orbit in which electron is revolving, m is mass of the electron, and r is the radius of the nth orbit).

**Which is momentum of a quantum wave particle? ›**

The point is that in quantum mechanics we interpret the wave number as being a measure of the momentum of a particle, with the rule that **p=ℏk**, so that relation (38.7) tells us that Δp≈h/Δx.

**Why does the momentum stay constant? ›**

The momentum of an object will never change if it is left alone. **If the 'm' value and the 'v' value remain the same, the momentum value will be constant**. The momentum of an object, or set of objects (system), remains the same if it is left alone. Within such a system, momentum is said to be conserved.

### Why momentum is never created or destroyed? ›

The conservation of momentum states that, within some problem domain, the amount of momentum remains constant; momentum is neither created nor destroyed, but **only changed through the action of forces as described by Newton's laws of motion**.

**What is the opposite of momentum? ›**

So, inertia describes an object's resistance to change in motion (or lack of motion), and momentum describes how much motion it has.

**Is momentum a scalar or vector? ›**

The momentum of a body is **a vector quantity**, for it is the product of mass, a scalar, by velocity, a vector.

**What causes momentum? ›**

It is **the product of an object's mass and velocity**. An object at rest has a momentum of 0. This helps distinguish momentum from inertia. An object must be in MOTION to have momentum.

**Is momentum a form of energy? ›**

**Momentum is a form of energy**. If an object has momentum, then it must also have mechanical energy. If an object does not have momentum, then it definitely does not have mechanical energy either.

**Do free electrons have momentum? ›**

**Electrons in free space can carry quantized orbital angular momentum (OAM) projected along the direction of propagation**. This orbital angular momentum corresponds to helical wavefronts, or, equivalently, a phase proportional to the azimuthal angle.

**How do we define momentum of a particle? ›**

momentum, **product of the mass of a particle and its velocity**. Momentum is a vector quantity; i.e., it has both magnitude and direction. Isaac Newton's second law of motion states that the time rate of change of momentum is equal to the force acting on the particle.

**Do all particles have momentum? ›**

Not only is momentum conserved in all realms of physics, but **all types of particles are found to have momentum**. We expect particles with mass to have momentum, but now we see that massless particles including photons also carry momentum.

**Is momentum a property of photon? ›**

It is now a well-established fact that **photons do have momentum**. In fact, photon momentum is suggested by the photoelectric effect, where photons knock electrons out of a substance.