What is the Schrödinger equation?

Schrödinger equation is a prominent equation in quantum mechanics. This equation is used to predict the behavior of quantum states or quantum particles. In quantum mechanics, the particles are treated as waves. A quantity called wave function is used to represent a quantum state. The wave function completely describes the quantum system. This equation can be used to study the particles like electrons, protons and, neutrons. It is also used to study the structure of hydrogen atoms and harmonic oscillators.

Wave function

Let us assume that an object of a certain mass is moving with a certain velocity. In classical mechanics, if we know the position and momentum of the object at some point in time then we can predict the position and momentum in the future. But it is not the case in quantum mechanics. In a quantum mechanical system, particles have wave-like nature.

 For the particles like electrons, protons, and photons the position and momentum cannot be observed simultaneously and accurately. This property is known as Heisenberg's uncertainty principle.  The quantum particles like electrons, photons obey this uncertainty rule. In quantum physics, it is impossible to get the exact position of any particles. Hence particles are treated as waves this property is also called wave-particle duality. A wave function gives complete information about the particle. The wave function is usually complex. The square of the absolute wave function gives the probability of finding a particle at some region of space.

The wave function is usually represented using the Greek letter $\psi$ or $\Psi$.

The probability density P is given as follows,

$P={\left|\psi (x,t)\right|}^2$

P is the probability of finding the particle in the coordinate x.

Operators

In quantum mechanics, the quantities like position, momentum, and energy can be assigned to operators. When a particular operator is operated with the wave function the quantity associated with the operator is obtained. The obtained quantity is called the eigenvalue. The operators are also called eigenoperators.

The operators are defined using complex functions. The complex functions are defined in Hilbert space. In Hilbert space, each state of the quantum system can be expressed using state vectors.

Consider a wave function $\psi (x,t)$, let $\hat{p}$  be the momentum operator. The momentum operator can be operated on the wavefunction as follows,

$\hat{p}\psi (x,t)=p\psi (x,y)$

The quantity p is called the eigen value of the position operator.

There are various quantum-mechanical operators like energy operator, position operator, and angular momentum operator, etc.

The momentum operator is defined as follows,

$\hat{p}=i\hslash \frac{\partial }{\partial x}$

The energy operator is defined as follows,

$\hat{H}=i\hslash \frac{\partial }{\partial t}$

$\hat{H}$ is called the Hamiltonian operator.

Time-dependent Schrödinger equation

The non-relativistic Schrodinger’s time-dependent equation for a single quantum particle of the wave function $\psi (x,t)$ is given as follows,

$i\hslash \frac{\partial }{\partial t}\psi (x,t)=\left[-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V\right]\psi (x,t)$

Where,

V is the potential energy exerted by the quantum state.

$\hslash$ is the reduced plank’s constant.

m is the mass of the particle.

This equation is analogous to Newton’s second law in classical mechanics. By knowing the initial momentum and position, future predictions can be made using Newton's law. In quantum mechanics, the Schrodinger equation is used to predict the future state of the quantum system.

Time independent Schrödinger equation

The Schrödinger equation can be expressed as time-independent form as shown below,

$\hat{H}\psi =E\psi$

Where, $\hat{H}$ is the Hamiltonian operator. This operator gives the total energy of the quantum system.

E is the energy eigenvalue. The equation can also be written as follows,

$E\psi =\left[\frac{p^2}{2m}+V\right]\psi$

In this equation, E is the total energy of the system and V is the potential energy. The term $\frac{p^2}{2m}$is the total kinetic energy. Hence the schrödinger equation is based on the conservation of energy.

$E\psi (x,t)=\left[-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V\right]\psi (x,t)$

$-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}\psi (x,t)=\left(E-V\right)\psi (x,t)$

$\frac{{\partial }^2}{\partial x^2}\psi (x,t)+\frac{2m}{{\hslash }^2}\left(E-V\right)\psi (x,t)=0$

This equation is known as the time-independent Schrödinger equation.

Particle in a one-dimensional box

One dimensional box is a simple system on which the Schrödinger equation can be applied to obtain the various state. Consider an electron of mass m in a one-dimensional box. In the one-dimensional box, the electron is free which means the net potential inside the box is zero. Let l be the length of the box.

If $x>l$ then $V=\infty$ (Outside the box)

If $x<0$ then $V=\infty$ (Outside the box)

Otherwise,

$V=0$ (Inside the box)

Hence, the particle cannot exist outside the box.

Let us apply the Schrödinger wave equation to the particle inside the box.

$-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}\psi (x,t)=E\psi (x,t)$

Which is a second-order differential equation. The solution of the above equation is as follows,

$\psi \left(x\right)=A\sin \left(kx\right)+B\cos \left(kx\right)$

Where, $k=\frac{\sqrt{2mE}}{\hslash }$

A and B are arbitrary constants.

To find constants A, B and k

When $x=0$,

$\psi \left(0\right)=0$

or $A\sin \left(0\right)+B\cos \left(0\right)=0$

or $B=0$      [Since  $\cos \left(0\right)=1$ and $\sin \left(0\right)=0$]

At x=l,

$\psi \left(l\right)=0=A\sin \left(kl\right)$

The term kl is a multiple of $\pi$ since $\sin \left(n\pi \right)=0$. Hence,

$k=\frac{n\mathrm{\pi }}{l}$      where $n=1,2,3,\dots$

Hence the wave function of the particle inside the box is given as,

$\psi (x,t)=A\sin \left(\frac{n\pi }{l}\right)$

Where A is the amplitude of the wave function.

Hence the energy eigenvalue becomes,

$E_n=\frac{n^2{\mathrm{\pi }}^2{\hslash }^2}{2m{l}^2}$

Where $E_n$ is the energy of the nth state.

Formulas

Schrodinger’s time-dependent wave equation is expressed as shown below,

$i\hslash \frac{\partial }{\partial t}\psi (x,t)=\left[-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V\right]\psi (x,t)$

Schrodinger’s time-independent wave equation is given as follows,

$\hat{H}\psi =E\psi$

or

$E\psi (x,t)=\left[-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V\right]\psi (x,t)$

Context and Applications

This topic is significant in physics for both undergraduate and graduate courses, especially for Masters in physics, and Bachelors in physics.

Practice Problems

Question 1: In quantum mechanics, it is impossible to measure the position and momentum _____.

(a) Accurately

(b) Simultaneously

(c) Both accurately and simultaneously

(d) Precisely

Answer: Option (c) is correct.

Explanation: According to Heisenberg’s uncertainty principle, the fundamental nature of a quantum system is that it is impossible to observe the position and momentum both accurately and simultaneously.  Hence option (c) is correct.

Question 2: Hamiltonian operator is a ____.

(a) Energy operator

(b) momentum operator

(c) Position operator

(d) Spin operator

Answer: Option (a) is correct.

Explanation: Hamiltonian operator is an energy operator. When it is operated on a wave function, total energy i.e. the sum of kinetic energy and potential energy is obtained as an eigenvalue.

Question 3: The square of the absolute value of a wave function gives the following quantity____.

(a) Energy

(b) Probability density

(c) Momentum

(d) Position

Answer: Option (b) is correct.

Explanation: The wave function is not physically significant. Bit the square of the absolute value of the wave function is nothing but the probability density. It indicates the region of space in which the particle is more likely to be found.

Question 4: For a free particle, _____ is zero.

(a) Kinetic energy

(b) Momentum

(c) Potential energy

(d) Angular momentum

Answer: Option (c) is correct.

Explanation: A particle is said to be a free particle when the net potential energy is zero. The free particle has maximum kinetic energy.

Question 5: Schrodinger equation is based on ____.

(a) Conservation of momentum

(b) Conservation of charge

(c) Conservation of energy

(d) Dirac equation

Answer: Option (d) is correct.

Explanation: In the Schrödinger equation, the Hamilton operator gives the total energy of the system. This operator is equated to the sum of kinetic energy and potential energy of the quantum system. Hence, conservation of energy is the correct answer.