SPA4402 Modern Physics. The wave function and the Schrodinger equation

SPA4402 Modern Physics

The wave function and

the Schrodinger equation

A free particle, by definition, is one that is not confined (that is, it can be located anywhere), it has a definite momentum,

wavelength, and energy.

A free particle, by definition, is one that is not confined (that is, it can be located anywhere), it has a definite momentum,

wavelength, and energy.

A confined particle is a wave packet (that is, likely to be located in a region of space, e.g. ∆𝑥 in one dimension). In quantum

mechanics we want to analyse confined particles, for example an electron confined to a specific atom.

Example: Electron confined by electron confined by a potential energy well.

The potential energy of the electron in the middle section is zero whereas the

electrical potential in the side sections is

− 𝑉₀

If the potential energy varies abruptly between the centre section and the side sections,

𝑈₀ = −𝑒 −𝑉₀ = +𝑒𝑉₀

The kinetic energy, 𝐾 , of an electron confined to the central section must be less than 𝑈₀.

The kinetic energy, 𝐾 , of an electron confined to the central section must be less than 𝑈₀.

If 𝐾 ≪ 𝑈₀, there is zero probability of finding the electron in the side sections because the confining potential is

discontinuous at points A and B.

This is physical situation is termed the infinite square well. In contrast to the free particle, only certain values of wavelength are allowed.

This statement follows from one which we made earlier:

This is physical situation is termed the infinite square well. In contrast to the free particle, only certain values of wavelength are allowed.

This statement follows from one which we made earlier:

The probability of finding a particle at any point depends on the amplitude of its de Broglie wave at that point.

Probability to observe particles ∝ 𝑑𝑒 𝐵𝑟𝑜𝑔𝑙𝑖𝑒 𝑤𝑎𝑣𝑒 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 ²

So the boundary conditions impose limits on the

wavelengths that can be

confined to the potential well.

This is another way of saying there is zero probability of

finding the electron in the side sections.

Consequently, the de Broglie relationship, 𝜆 = ^ℎ

𝑝 , tells us that only certain values of momentum are allowed.

Consequently, the de Broglie relationship, 𝜆 = ^ℎ

𝑝 , tells us that only certain values of momentum are allowed.

This means that energy is not free to take any value, instead energy is restricted to discrete values. This is known as quantisation of energy.

Consequently, the de Broglie relationship, 𝜆 = ^ℎ

𝑝 , tells us that only certain values of momentum are allowed.

This means that energy is not free to take any value, instead energy is restricted to discrete values. This is known as quantisation of energy.

From the image above we can see that the wavelength of a particle in the centre section is restricted to

𝜆_𝑛 = 2𝐿

𝑛 , 𝑛 = 1,2,3 …

Identical to the set of wavelengths for classical standing waves on a string between two points but remember there is no ‘medium’ for de Broglie waves. From the de Broglie relationship,

𝑝_𝑛 = 𝑛 ℎ 2𝐿

Identical to the set of wavelengths for classical standing waves on a string between two points but remember there is no ‘medium’ for de Broglie waves. From the de Broglie relationship,

𝑝_𝑛 = 𝑛 ℎ 2𝐿

Since the energy of the particle in the centre section, is only kinetic energy, 𝐾 = 𝑝²/2𝑚

𝐸_𝑛 = 𝑛² ℎ² 8𝑚𝐿²

Quantisation of energy accompanies every attempt to confine a particle to a region of space. It is one of the principle features of quantum theory.

Remember that the uncertainty principle applied to a wave packet, ∆𝑝_𝑥 = 𝑝_𝑥² _av − (𝑝_{𝑥,𝑎𝑣})².

Remember that the uncertainty principle applied to a wave packet, ∆𝑝_𝑥 = 𝑝_𝑥² _av − (𝑝_{𝑥,𝑎𝑣})².

Here there is an equal probability of the electron in the

central section of the moving from left to right and right to left, so 𝑝_{𝑥,𝑎𝑣} = 0 and ∆𝑝_𝑥² = 𝑝_𝑥² _av.

Remember that the uncertainty principle applied to a wave packet, ∆𝑝_𝑥 = 𝑝_𝑥² _av − (𝑝_{𝑥,𝑎𝑣})².

Here there is an equal probability of the electron in the

central section of the moving from left to right and right to left, so 𝑝_{𝑥,𝑎𝑣} = 0 and ∆𝑝_𝑥² = 𝑝_𝑥² _av. Consequently,

∆𝑥∆𝑝_𝑥~𝐿𝑛 ℎ

2𝐿 ~𝑛ℎ > ℏ 2

The result of confining the particle is consistent with the Heisenberg uncertainty relationship for location and

momentum, ∆𝑥∆𝑝_𝑥 ≥ ^ℏ

2

In quantum mechanics, the differential equation whose solution gives us the wave behaviour of particles is the

Schrödinger equation. It cannot be derived from any previous laws or postulates, its correctness can be tested by comparing predictions with experimental results.

The Schrödinger equation gives results which correctly account for non-relativistic motion at the atomic and subatomic level.

T^HE WAVE FUNCTION

The Schrödinger equation uses a wave function to describe the state of a particle, the customary symbol is Ψ or 𝜓 .

The Schrödinger equation uses a wave function to describe the state of a particle, the customary symbol is Ψ or 𝜓 .

- In general, Ψ is a function of all space coordinates and time, whereas 𝜓 is a function of only space coordinates.

The Schrödinger equation uses a wave function to describe the state of a particle, the customary symbol is Ψ or 𝜓 .

- In general, Ψ is a function of all space coordinates and time, whereas 𝜓 is a function of only space coordinates.

- Ψ may be complex, i.e. with real and imaginary parts (the imaginary part is a real function multiplied by the imaginary number 𝑖 = −1).

The Schrödinger equation uses a wave function to describe the state of a particle, the customary symbol is Ψ or 𝜓 .

- In general, Ψ is a function of all space coordinates and time, whereas 𝜓 is a function of only space coordinates.

- Ψ may be complex, i.e. with real and imaginary parts (the imaginary part is a real function multiplied by the imaginary number 𝑖 = −1).

- The wave function describes the particle, but we cannot define the function itself in terms of anything material.

- Just as the wave function 𝑦(𝑥, 𝑡) for mechanical waves on a string provides a complete description of the motion, so the wave function Ψ(𝑥, 𝑦, 𝑧, 𝑡) for a particle contains all the information that can be known about a particle.

- Just as the wave function 𝑦(𝑥, 𝑡) for mechanical waves on a string provides a complete description of the motion, so the wave function Ψ(𝑥, 𝑦, 𝑧, 𝑡) for a particle contains all the information that can be known about a particle.

- We can only describe how it is related to physically observable effects.

- Just as the wave function 𝑦(𝑥, 𝑡) for mechanical waves on a string provides a complete description of the motion, so the wave function Ψ(𝑥, 𝑦, 𝑧, 𝑡) for a particle contains all the information that can be known about a particle.

- We can only describe how it is related to physically observable effects.

- Quantum mechanics is the mathematical theory that describes how to use Ψ(𝑥, 𝑦, 𝑧, 𝑡) to determine average values of the particle’s position, velocity, momentum, energy and angular momentum.

- Just as the wave function 𝑦(𝑥, 𝑡) for mechanical waves on a string provides a complete description of the motion, so the wave function Ψ(𝑥, 𝑦, 𝑧, 𝑡) for a particle contains all the information that can be known about a particle.

- We can only describe how it is related to physically observable effects.

- Quantum mechanics is the mathematical theory that describes how to use Ψ(𝑥, 𝑦, 𝑧, 𝑡) to determine average values of the particle’s position, velocity, momentum, energy and angular momentum.

- Born interpretation of the wave function: Ψ(𝑥, 𝑦, 𝑧, 𝑡) ²𝑑𝑉 is the

probability that the particle will be found at time 𝑡 within a volume 𝑑𝑉 around point (𝑥, 𝑦, 𝑧).

This probability density (probability per unit length) for a time- independent wave function in one dimension is expressed as,

𝑃(𝑥)𝑑𝑥 = 𝜓(𝑥) ²𝑑𝑥

- The particle must be located somewhere between 𝑥 = −∞ and 𝑥 = +∞ so,

𝜓(𝑥) ²𝑑𝑥 = 1

+∞

−∞

This is the normalisation condition.

Generally, Ψ(𝑥, 𝑦, 𝑧, 𝑡) ²at a particular point varies with time STATIONARY STATES

Generally, Ψ(𝑥, 𝑦, 𝑧, 𝑡) ²at a particular point varies with time

If a particle is in a state with definite energy, such as a confined electron, its value at a particular point is

independent of time.

STATIONARY STATES

Generally, Ψ(𝑥, 𝑦, 𝑧, 𝑡) ²at a particular point varies with time

If a particle is in a state with definite energy, such as a confined electron, its value at a particular point is

independent of time.

The condition Ψ ²is independent of time is called a stationary state, these are very important in quantum

mechanics because for every definite-energy stationary state there is a specific wave function.

STATIONARY STATES

An important result from quantum mechanics states: For a particle in a state of definite energy, the time-dependent wave function

Ψ(𝑥, 𝑦, 𝑧, 𝑡)can be written as a product of the time-independent wave function and a simple function of time,

Ψ 𝑥, 𝑦, 𝑧, 𝑡 = 𝜓 𝑥, 𝑦, 𝑧 𝑒^{−𝑖𝐸𝑡/ℏ}

This is the time-dependent wave function for a stationary state. It is a complex function, the exponent is defined by Euler’s formula:

𝑒^−𝑖𝜃 = cos𝜃 + 𝑖sin𝜃. For complex functions, Ψ ²= Ψ. Ψ^∗

This is the time-dependent wave function for a stationary state. It is a complex function, the exponent is defined by Euler’s formula:

𝑒^−𝑖𝜃 = cos𝜃 + 𝑖sin𝜃. For complex functions, Ψ ²= Ψ. Ψ^∗

where Ψ^∗ is the complex conjugate, which is found by replacing all 𝑖in a complex number with –𝑖, so

Ψ^∗ 𝑥, 𝑦, 𝑧, 𝑡 = 𝜓^∗ 𝑥, 𝑦, 𝑧 𝑒^{+𝑖𝐸𝑡/ℏ}

Hence,

Ψ 𝑥, 𝑦, 𝑧, 𝑡 ² = Ψ^∗ 𝑥, 𝑦, 𝑧, 𝑡 . Ψ 𝑥, 𝑦, 𝑧, 𝑡

= 𝜓 𝑥, 𝑦, 𝑧 𝑒^{−𝑖𝐸𝑡/ℏ}.𝜓^∗ 𝑥, 𝑦, 𝑧 𝑒^{+𝑖𝐸𝑡/ℏ}

= 𝜓 𝑥, 𝑦, 𝑧 .𝜓^∗ 𝑥, 𝑦, 𝑧

= 𝜓 𝑥, 𝑦, 𝑧 ²

This justifies use of the term ‘stationary state’ for a state of definite energy

To justify the form of the Schrödinger equation:

Consider a free particle in one dimension, its wave function is likely to be that of a simple de Broglie wave such as:

𝜓 𝑥 = 𝐴. sin 𝑘𝑥,

where 𝐴 is the amplitude and 𝑘 = 2𝜋/𝜆 is the wave number.

(We have used 𝜓 𝑥 here because Ψ ²is time-dependent, i.e. we are not considering a travelling wave)

T^HE SCHRÖDINGER EQUATION

Now consider its first and second derivative,

𝑑𝜓

𝑑𝑥 = 𝑘𝐴. cos 𝑘𝑥, 𝑑²𝜓

𝑑𝑥² = −𝑘²𝐴. sin 𝑘𝑥 = −𝑘²𝜓 𝑥

The second derivative gives the original function again.

Since non-relativistic kinetic energy, 𝐾 = 𝑝²

2𝑚 = (ℎ/𝜆)²

2𝑚 = ℏ²𝑘² 2𝑚 We can write,

𝑑²𝜓

𝑑𝑥² = −𝑘²𝜓 𝑥 = − 2𝑚

ℏ² (𝐸 − 𝑈)𝜓 𝑥

𝐸 = 𝐾 + 𝑈 is the non-relativistic total energy of the particle (𝑈is potential energy).

For a free particle 𝑈 = 0 so 𝐸 = 𝐾 but generally, the above equation becomes,

− ℏ² 2𝑚

𝑑²𝜓

𝑑𝑥² + 𝑈(𝑥)𝜓 𝑥 = 𝐸𝜓 𝑥

This is the time-independent Schrödinger equation (TISE) for one- dimensional motion.

− ℏ² 2𝑚

𝑑²𝜓

𝑑𝑥² + 𝑈(𝑥)𝜓 𝑥 = 𝐸𝜓 𝑥

This is the time-independent Schrödinger equation (TISE) for one- dimensional motion.

The Schrodinger equation plays the sample role in quantum mechanics as do Newton’s laws in mechanics and Maxwell’s equations in

electromagnetism.

− ℏ² 2𝑚

𝑑²𝜓

𝑑𝑥² + 𝑈(𝑥)𝜓 𝑥 = 𝐸𝜓 𝑥

This is the time-independent Schrödinger equation (TISE) for one- dimensional motion.

The TISE plays the sample role in quantum mechanics as do Newton’s laws in mechanics and Maxwell’s equations in electromagnetism.

Our understanding of atoms, molecules, atomic nuclei, and electrons in solids is based on solutions to this equation for that system.

To solve the TISE, we assume that we know 𝑈(𝑥) and obtain the wave function 𝜓 𝑥 and energy for that potential energy.

We find that it is possible to obtain solutions to the equation only for particular values of 𝐸 , which are known as energy eigenvalues.

Since the Schrödinger equation is linear, any constant multiplying a solution is also a solution.

In quantum mechanics we are dealing with probabilities rather than certainties so the outcome of a single measurement of a physical

quantity cannot be guaranteed. For example, the number of times we measure position 𝑥 is proportional to the probability 𝑃(𝑥)𝑑𝑥 to find the particle in the interval 𝑑𝑥 at 𝑥 so a the average value of extracted from multiple measurements is,

𝑥_av = ׬_−∞^+∞ 𝑃(𝑥)𝑥𝑑𝑥

׬_−∞^+∞ 𝑃(𝑥)𝑑𝑥 =

−∞

+∞

𝜓(𝑥) ²𝑥𝑑𝑥

This follows from ׬_−∞^+∞ 𝑃(𝑥)𝑑𝑥 = 1 when the normalisation condition is applied.

By analogy, the average value of any function of can be found from,

𝑓 𝑥 _av =

−∞

+∞

𝑃(𝑥)𝑓(𝑥)𝑑𝑥 =

−∞

+∞

𝜓(𝑥) ²𝑓(𝑥)𝑑𝑥

These average values are called expectation values.

Example: The free particle

First assume a constant potential energy, 𝑈 𝑥 = 0, so the TISE becomes,

− ℏ² 2𝑚

𝑑²𝜓

𝑑𝑥² = 𝐸𝜓 𝑥 The general solution to this equation has the form,

𝜓 𝑥 = 𝐴. sin 𝑘𝑥 + 𝐵. cos(𝑘𝑥) where 𝑘 = ^2𝜋

𝜆 .

And the energy of the particle is,

APPLICATIONS OF THE SCHRÖDINGER EQUATION

𝐸 = ℏ²𝑘² 2𝑚

This is what we would have expected from a de Broglie wave with momentum 𝑝 = ^ℎ

𝜆 , 𝐸 is all kinetic energy, 𝐾 = ^𝑝2

2𝑚. Wave vector 𝑘 = ^2𝜋

𝜆 can take any value so energy is not quantized.

𝐸 = ℏ²𝑘² 2𝑚

Example: The infinite potential energy well.

Express the potential energy as,

𝑈 𝑥 = 0 0 ≤ 𝑥 ≤ 𝐿 𝑈 𝑥 = ∞ 𝑥 < 0, 𝑥 > 𝐿

For the TISE to be meaningful when 𝑈 𝑥 → ∞ , 𝜓 = 0 so that 𝑈𝜓 is finite so, 𝜓 𝑥 = 0 𝑥 < 0, 𝑥 > 𝐿

For 0 ≤ 𝑥 ≤ 𝐿, when 𝑈 𝑥 = 0, the solution to the TISE will be the same as the free particle solution:

𝜓 𝑥 = 𝐴. sin 𝑘𝑥 + 𝐵. cos 𝑘𝑥 0 ≤ 𝑥 ≤ 𝐿 with,

𝑘 = ^2𝑚𝐸

ℏ² .

To find 𝐴 and 𝐵, we must consider continuity of 𝜓 𝑥 across the 𝑈 = 0 to 𝑈 = ∞ boundaries. This is called applying the boundary conditions,

𝜓 0 = 0 = 𝐴. sin 0 + 𝐵. cos 0 𝑥 = 0, which gives 𝐵 = 0, so

𝜓 𝐿 = 0 = 𝐴. sin 𝑘𝐿 𝑥 < 𝐿

This means either 𝐴 = 0 (which means the particle doesn’t exist) or sin 𝑘𝐿 = 0, which is only true when

𝑘𝐿 = 𝑛𝜋 𝑛 = 1,2,3 …

To find 𝐴 and 𝐵, we must consider continuity of 𝜓 𝑥 across the 𝑈 = 0 to 𝑈 = ∞ boundaries. This is called applying the boundary conditions,

𝜓 0 = 0 = 𝐴. sin 0 + 𝐵. cos 0 𝑥 = 0, which gives 𝐵 = 0, so

𝜓 𝐿 = 0 = 𝐴. sin 𝑘𝐿 𝑥 > 𝐿

This means either 𝐴 = 0 (which means the particle doesn’t exist) or sin 𝑘𝐿 = 0, which is only true when

𝑘𝐿 = 𝑛𝜋 𝑛 = 1,2,3 … Since 𝑘 = ^2𝑚𝐸

ℏ² ,

𝐸_𝑛 = ℎ²𝑛²

8𝑚𝐿² 𝑛 = 1,2,3 …

In other words, the solution to the TISE for a particle trapped in an infinite potential

energy well is a series of standing de Broglie waves.

Only certain values of 𝑘 are permitted so energy is quantised.

Or, 𝐸_𝑛 = 𝑛²𝐸₀, where 𝐸₀ is the ground state energy (for which 𝑛 = 1)

and the higher energy stationary (𝑛 > 1) states are referred to as excited states