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### How to Find the Hermitian Adjoint

In quantum physics, you’ll often work with Hermitian adjoints. The *Hermitian adjoint* — also called the *adjoint* or *Hermitian conjugate* — of an operator A is denoted [more…]

### How to Arrange Eingenvectors

In quantum physics, the eigenvectors of a Hermitian operator define a complete set of orthonormal vectors — that is, a complete basis for the state space. When viewed in this “eigenbasis,” which is built [more…]

### How to Find the Eigenvectors and Eigenvalues of an Operator

In quantum physics, if you’re given an operator in matrix form, you can find its eigenvectors and eigenvalues. For example, say you need to solve the following equation: [more…]

### How Pair Production and Pair Annihilation Define Light Particles

By observing both pair production and pair annihilation, 20th-century physicists were able to prove that light has the characteristics of a particle. This process of discovery began in 1928, when the physicist [more…]

### How de Broglie Showed that All Particles Display Wave-Like Properties

In 1923, the physicist Louis de Broglie suggested that not only did waves exhibit particle-like aspects, but that the reverse was also true — all material particles should display wave-like properties. [more…]

### How to Use the Heisenberg Uncertainty Principle in Relation to Position and Momentum

Quantum physicists understand that matter exhibits wave-like properties, which means that matter, like waves, aren't localized in space. This fact inspired Werner Heisenberg, in 1927, to come up with his [more…]

### How Quantum Physics Converts Momentum and Position into Probabilities

Quantum physics, unlike classical physics, is completely nondeterministic. You can never know the *precise* position and momentum of a particle at any one time. You can give only probabilities for these [more…]

### How to Assemble Relative Probabilities into a Vector

In quantum physics, probabilities take the place of absolute measurements. Say you've been experimenting with rolling a pair of dice and are trying to figure the relative probability that the dice will [more…]

### How to Use Ket Notation in Quantum Physics

In quantum physics, ket notation makes the math easier than it is in matrix form because you can take advantage of a few mathematical relationships. For example, here’s the so-called Schwarz inequality [more…]

### How the Compton Effect of Light Explains Wavelength Shift

Although Max Planck and Albert Einstein postulated that light could behave as both a wave and a particle, it was Arthur Compton who finally proved that this was possible. His experiment involved scattering [more…]

### How to Use Kets, the Hermitian Conjugate, and Bra-ket Notation

What do Dirac notation and the Hermitian conjugate have in common? They help physicists to describe really, really big vectors. In most quantum physics problems, the vectors can be infinitely large — for [more…]

### How to Use Operators for Quantities in Quantum Physics

In quantum physics, you can use operators to extend the capabilities of bras and kets. Although they have intimidating-sounding names like Hamiltonian, unity, gradient, linear momentum, and Laplacian, [more…]

### How to Use Linear Operators in Quantum Physics

In quantum physics, you need to know how to use linear operators. An operator A is said to be *linear* if it meets the following condition: [more…]

### How Quantum Physicists Describe a Black Body

One of the major ideas of quantum physics is *quantization*— measuring quantities in discrete, not continuous, units. The idea of quantized energies arose with one of the earliest challenges to classical [more…]

### How Physicists Solved the Photoelectric Effect of Light

The photoelectric effect was one of many experimental results that made up a crisis for classical physics around the turn of the 20th century. It was also one of Einstein’s first successes, and it provides [more…]

### How to Find the Commutator of Operators

In quantum physics, the measure of how different it is to apply operator A and then B, versus B and then A, is called the operators’ *commutator*. Here’s how you define the commutator of operators A and [more…]

### How to Find the Heisenberg Uncertainty Relation from Scratch

If you’ve read through the last few sections, you’re now armed with all this new technology: Hermitian operators and commutators. How can you put it to work? You can come up with the Heisenberg uncertainty [more…]

### How to Work with Eigenvectors and Eingenvalues

In quantum physics, when working with kets, it is useful to know how to use eigenvectors and eigenvalues. Applying an operator to a ket can result in a new ket: [more…]

### How to Find the Inverse of a Large Matrix

Finding the inverse of a large matrix often isn’t easy, so quantum physics calculations are sometimes limited to working with unitary operators, U, where the operator’s inverse is equal to its adjoint, [more…]

### How to Derive the Schrödinger Equation

In quantum physics, the Schrödinger technique, which involves wave mechanics, uses wave functions, mostly in the position basis, to reduce questions in quantum physics to a differential equation. [more…]

### How to Find the Expectation Value of a Dice Roll

Given that everything in quantum physics is done in terms of probabilities, making predictions becomes very important. And the biggest such prediction is the expectation value. The expectation value of [more…]