Quantum Physics

The Schrödinger equation: Introduction

Cont’d from “The Hamiltonian operator: Introduction”, see “Operators

In its most general sense, the Schrödinger equation is simply the eigen-equation for the Hamiltonian acting on a wavefunction | ψ ⟩, given by

Now, by applying the Hamiltonian operator to the wavefunction, it becomes

which we can substitute into its eigen-equation to obtain

where ∇2 is the Laplacian operator. This is known as the time-independent Schrödinger equation (TISE) in 3-dimensions – often introduced as the linear TISE:

i.e. the TISE in 1-dimension.

Note how in both cases the potential energy is still in operator form, which can be varied depending on the scenario. For example, charged particles will have potential energy provided by the Coulomb potential

It is rare that the potential energy will take the form of a conventional operator, leading to the energy conservation form of the Hamiltonian being written simply as

though this is only a notational difference.

Time dependency

Now, to examine the time dependence of the Schrödinger equation we must apply the Hamiltonian operator to a time dependent wavefunction, | ψ(r, t) ⟩. We find its eigen-equation to be

which leads to the time-dependent Schrödinger equation (TDSE)

or in 1-dimension

Solving this equation can tell us information about the time evolution of the wavefunction. This means we can find how it moves or how it interacts within the confines of quantum mechanics.

It is worth noting that the Schrödinger equation often sees a correction in the mass of the particle m in scenarios of interaction between two or more particles. This is best expressed by the reduced mass μ.

The universe is a song, singing itself.

No, really. The solutions of the Schrödinger Equation are harmonics, just like musical notes.

Quantum state of an electron orbiting a hydrogen atom where n=6, l=4, and m=1 (spin doesn’t matter):

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This is an equal superposition of the |3,2,1> and |3,1,-1> eigenstates:

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This is an equal superposition of the |3,2,2> and |3,1,-1> eigenstates:

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This is an equal superposition of the |4,3,3> and |4,1,0> eigenstates:

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What is an eigenstate? It’s a convenient state to use as a basis. We get to decide which quantum states are “pure” and which “mixed”. There’s an easy way and a hard way; the easy way is to use eigenstates as the pure states.

More mathematically: the Schrödinger equation tells us what’s going on in an atom. The answers to the Schrödinger equation are complex and hard to compare. But phrasing the answers as combinations of eigenfunctions makes them comparable. One atom is 30% state A, 15% state B, 22% state C … and another atom is different percentages of the same states.

Just like vectors in 3-D space, where you can orient the axes differently – you can pick different directions for x, y, and z to point in. But now the vectors are abstract, representing states. Still addable so still vectors. Convex or linear combinations of those “pure” states describe the “mixed” states.


SOURCE: Atom in a Box

Example of a multifractal electronic eigenstate at the Anderson localization transition in a system with 1367631 atoms. Each cube indicates by its size the probability to find the electron at the given position. The color scale denotes the position of the cubes along the axis into the plane.

By RudoRoemer (GFDL or CC BY-SA 3.0), via Wikimedia Commons

50 Years of Anderson Localization, Short HistorySympossium, Institut Henri Poincaré, Paris, December 4-5, 2008

Quantum Physics

Operators in quantum mechanics

Cont’d from “Dirac notation: Analogy with Cartesian vectors”, see “Dirac notation: Introduction

Throughout mathematics, an operator is defined as:

“A mapping of a function to produce a new function.”

A common example of a regular operator is the differential operator.

Quantum operators are defined by the first postulate of quantum mechanics:

”For any well-defined physical quantity (such as momentum or energy), there exists a quantum mechanical operator (called an “observable”) that yields a value which is an eigenvalue of the function the operator acts upon.”

In quantum physics, operators are written using a “hat”, above the symbol.

Therefore, an observable Ô acting of a function ψ yields a measurement o, which is an eigenvalue of ψ. This is given by the equation

or in Dirac notation

Operators in matrix form

In a discrete QM system, an operator is defined to be a Hermitian matrix which has an eigenstate vector.

Hermitian matrices

In matrix mechanics, a Hermitian matrix is one which has its conjugate transpose equal to the original matrix.

Thus, it must be a square matrix with symmetric elements with equal real parts but oppositely-signed imaginary parts. Thus a general c × c Hermitian matrix ℍ is

where h ∈ ℂ. More explicitly this is

Common operators

Some common operators in quantum mechanics are:


Linear momentum:

where = (/x, /y, /z) is the del (or nabla) operator, i is the imaginary unit and ℏ = h∕2π is the reduced Planck’s constant. Alternatively, in 1-dimension this is

where /x is the x-component of the del operator.

Kinetic energy:

Using the momentum operator we defined previously,

and since i2 ≡-1, the kinetic energy operator becomes

where ∇2 = (2/x2 + 2/y2 + 2/z2) is the Laplacian operator (derived from finding the divergence of del).