2006-7 Module 113 - Lecture 6

Module 113- Quantum theory and atomic spectroscopy LECTURE 6- The Schrödinger equation The idea that material objects have wave-like behaviour has many startling consequences, one of which is that waves undergo interference effects. Interference is the result of superposition of waves, and for quantum objects we find that the concept of superposition of wavefunctions is an important part of microscopic reality (we will see this in action in the last lecture). One man who did more than any to unravel the meaning of wave-particle duality was Erwin Schrödinger who demonstrated that the hydrogen atom could be understood clearly in terms of wavefunctions. It was also Schrödinger who later said that the superposition of wavefunctions was the only real mystery in quantum mechanics, and indeed will meet it many times in the coming years. Schrödinger equation and the operator formalism of quantum mechanics Schrödinger’s additional postulate was his wave-equation, which today we represent in the deceptively simple form

Hˆ Ψ = EΨ

(6.1)

With this equation, Schrödinger was proposing a new way of defining microscopic bodies. No longer shall quantum objects (electrons, atoms etc) be described as either waves or particles, they can be expressed as the solutions to the wave-equation, which we call a wavefunction, Ψ This is just a mathematical function, such as sin (x) etc, which can be used to define the object in a precise mathematical form. For comparison, think of how you would describe a wave (as a function like sin (x) which has a periodicity) or a particle (by its position in space or a vector). The wavefunction is simply a generalisation of the old mathematical descriptions.

Hˆ is the Hamiltonian, and is an example of an operator. An operator is a mathematical function that you must

apply to the wavefunction that follows it. This sounds more confusing than it really is. As an example, consider the differential

dy d (y) = dx dx d on the function y. In much the same way, the operator Ĥ dx acts on the wavefunction. We shall identify operators by capital letters with a carat (hat) on top e.g. pˆ is the

Here we can think of the operator as the operation momentum operator. A little bit of calculus

I know from previous years that many of you won’t have a clue what that above operator was, but you might have come across calculus before. It expresses the rate of change of one variable with respect to the change in another. A common example of this is recording the time it takes an athlete to cover 100m- in effect, you are measuring the change in his position with respect to a change in the local time. However, this can also be defined as the athlete’s velocity. If distance is s and time t then

ds = velocity dt

Mathematical functions like y = 2x + 1 can also be differentiated just like s in the above example. The result is (almost) always another function. There are in fact a series of rules about how to differentiate functions, and these are usually just memorised. For example

( )

d xn dx

= nx n − 1

for example

( )

d x3 dx

= 3x 2

(6.2)

of course, you can keep on taking differentials till you run out of functions and are left with just a constant (the differential of a constant is just 0). For example, the differential of velocity is the acceleration. In shorthand, differentiating twice is represented by

d2 dx 2 Some functions have special properties under the operation of

d . These functions are often listed in textbooks dx

and below I give an important example:

( )

d e ax dx

= ae ax ,

where a is a constant

(6.3)

The differential now has the same form as the original function, except for being multiplied by the constant a. If we differentiate again we find

( )

d2 e ax 2 dx

= a 2 e ax

(6.4)

The eigenvalue problem Now some wavefunctions and operators have a very special relationship, which we will illustrate now with an example. Let’s take a wavefunction

Ψ = e ax . d . We find Next, let’s apply the operator Aˆ = dx

⎛d ⎞ Aˆ Ψ = ⎜ ⎟e ax = ae ax ⎝ dx ⎠

(6.5)

This result is clearly the original wavefunction multiplied by a constant. Any equation of the form

Aˆ Ψ = aΨ

(6.6)

is known as an eigenvalue problem. A wavefunction that obeys such an equation is called an eigenfunction and a is its associated eigenvalue. Clearly the Schrödinger equation is an example of an eigenvalue problem, but the eigenvalue in question has a special significance- it is the energy E of the wavefunction. The Schrödinger equation is therefore a method of calculating the energy of a quantum object. The Hamiltonian is consequently the operator required to determine the energy of a wavefunction.

Schrödinger has replaced objects, such as electrons, with their mathematical descriptions, called wavefunctions, in order to find the properties of those objects, such as their momentum and by using his famous formula

Hˆ Ψ = EΨ we can find their energies. In general, we use the formula above to determine the wavefunction of an object or physical system, such as a hydrogen atom or a harmonic oscillator or anything we want. All we need (in theory) is not love but the Hamiltonian. This is pretty awesome stuff- turning two hundred years of physics on its head with a completely fresh description of nature. Yet, ultimately the Schrödinger equation will prove the quantization of energy levels in a hydrogen atom, which is after all what we are desperate to understand.

15. Solving the Schrödinger equation The best way to understand how the Schrödinger equation works is to try an example. Unfortunately, the only way to appreciate it is to dip into the mathematics. Don’t worry though: we will look at just two examples at this full mathematical level and both are relatively easy. My policy is to only do the mathematics when it will help and deepen your understanding of quantum theory and not just for fun (I am not that mad). The examples are 1) a free particle moving in one-dimension, such as an electron in the absence of an electric or magnetic field 2) a particle confined to a ring. Both these examples reveal fundamental features of quantum theory that help explain atomic structure! The Hamiltonian is obviously an operator, but we have said nothing so far about its mathematical form. This is simply because it depends somewhat on the physical characteristics of the quantum object and its external environment (hence, if you like, the forces acting upon it). The Hamiltonian of an electron in the absence of a magnetic field is different from one where a field is present, and equally will be different if we replace the electron by a neutron (because the neutron has no permanent electric charge). Example: the Schrödinger equation for a free body (one-dimensional motion only) There are many of these textbook cases of the Schrödinger equation but this is perhaps the easiest. You will find that writing down the Hamiltonian needed and the Schrödinger equation for a particular problem will not be that difficult, but with the Schrödinger equation the devil is in the (mathematical) detail required to solve it. Only very simple problems can be solved without resorting to mathematical techniques and computer hardware beyond the scope of our course. The example I have chosen is an uncharged object of mass m moving in one dimension (x). For the sake of argument, we will imagine a microscopic ball of dimensions typical for an atom of uniform composition (so we don’t have to worry about its internal structure). Classically, the energy would be given by

E =

p2 + V (x ) 2m

(6.7)

Where V ( x ) is the potential energy of the body, and

p2 the kinetic energy that it has. We require the 2m

corresponding Hamiltonian equation for this system. We find that it too can be clearly separated into two operators, one for the kinetic energy

−

h2 d 2 2m dx 2

h =

h 2π

(6.8)

And one for the potential energy with the resulting full Hamiltonian

h2 d 2 Hˆ = − + V (x ) 2m dx 2

(6.9)

The Schrödinger equation allows us to determine what wavefunction describes our microscopic ball with an energy operator given by eqn. (6.9). For the moment, we will set the potential energy to zero. This implies there are no external forces acting on our microscopic ball (we say it is a free body). The resulting Schrödinger equation is

⎛ h2 d 2 ⎞ ⎟ Ψ = EΨ Hˆ Ψ = −⎜⎜ 2 ⎟ m 2 dx ⎝ ⎠ Rearranging the above formula we get

d 2Ψ 2mE = − Ψ 2 dx h2

for a free body

(6.10)

d2 Obviously, this is an eigenvalue problem where the operator is simply 2 . Plundering the vast human knowledge dx on mathematical functions, we know that

d 2 e ax = a 2 e ax dx 2

(6.11)

Therefore, the function e ax is an eigenfunction for an operator of the type

Ψ = e ax

d2 . Therefore, dx 2 (6.12)

But what is a? Well, by comparing eqn. (6.11) with eqn. (6.10) we see that

a2 = −

2mE h2

(6.13)

But there is a nasty surprise here since m > 0 and E > 0, then

a2 < 0 This means that a must be a complex number. You have probably all met complex numbers before and wondered what possible use they could be: well, in quantum mechanics they are essential. We can therefore express a as a real number k multiplied by i, which is defined as the square root of minus 1

a = ik

i =

− 1

(6.14)

We can now find k by comparing both sides of equation (6.12)

a 2 = i 2 k 2 = −k 2 = −

2mE h2

k =

2mE h2

(6.15)

k is called the wavevector. We have found one solution to equation (6.10), namely

Ψ = e ikx The wavefunction of the free-body is just a mathematical function. However, we find that

Ψ = e −ikx is also a solution. From equation (6.10) alone there is no way to decide which is the better description of our quantum free-body, so instead we have to write a general solution

Ψ = Ae ikx + Be −ikx

(6.16)

This happens a lot in quantum mechanics- the values of A and B depend on any additional details of the wavefunction. Why did I use this example of the Schrödinger equation? Firstly, we saw that complex numbers play an important role in quantum mechanics. Secondly, and more crucially, it demonstrates that wave-particle duality alone does not automatically lead to quantization of energy. The electron has wave and particle properties, but there are an infinite number of possible solutions (with different values of energy E) of the 1-D free-body Schrödinger equation. The important thing to note is that the energy of the free-body is not quantized. In math-speak we say that wave-particle duality is a necessary but not sufficient condition for quantization. Impress grandma with that one. However we will see later how the physical reality of a wavefunction often leads to restrictions on the actual form a wavefunction can take, and therefore on the energy a particle can have (quantization!) In order to do this, we must understand a little more what a wavefunction actually represents.

16. The Born Interpretation of the wavefunction We have all the machinery (the Schrödinger equation and the HUP) necessary to reveal the quantum nature of microscopic objects, but is that all we need to consider? Is it just a “simple” case of solving the Schrödinger equation? Well no it’s not really. The wavefunction is a mathematical description of a quantum object. But it also must have a physical meaning, as electrons, for example, are real material objects. What is that meaning? Consider the kinetic energy of a free body (i.e. there are no external forces acting upon it)

E=

p2 2m

Inserting p =

h an rearranging we get λ db

λ db =

h

(6.17)

2mE

We see that if the kinetic energy is large, then λ db is small and vice versa. The wavelength of a free-body is longer the less kinetic energy it has. But from HUP we know that we really can’t talk about a monochromatic wavelength when talking about a localised particle so we need a more general description applicable to all (i.e. to waves, wavepackets and such like). This is the curvature of a wavefunction. A wave with high curvature corresponds to a short wavelength because it rapidly changes over a short distance. A high curvature therefore corresponds to

a high kinetic energy, which is rather nice and intuitive. Now consider the wavefunction below (you may meet this in a few weeks!)

For curve 1, the curvature is smaller than curve 2, indicating it has less kinetic energy. Curve 3 has the largest curvature of all so we know that this wavefunction corresponds to the highest kinetic energy. The wavefunction, in short, provides information concerning the kinetic energy of a quantum object. Is that all the wavefunction can tell us? Luckily, no, thanks to an insight by Max BORN. He made use of the similarity of the wavefunction to the wave theory of light. Classically, a light wave has amplitude, which represents the electric field. The square of the electric field, and hence the amplitude, is the intensity. Born reasoned that he could generalise this idea and talk about probabilities. The Born Interpretation of the wavefunction states that the square of the wavefunction at any point is proportional to the probability of finding the body at that point. Please note that it is proportional, not equal. The amplitude of a wavefunction at a point is often referred to as the probability amplitude. High amplitude means a high probability to find the body at that point. However, we have to be careful how we square our wavefunctions. Why? Well, more often than not the wavefunction will contain complex numbers, and they have a peculiar quality when they are squared. In general, we represent a complex number by the form A + iB That is, it has a real part and a complex part. Now, if I square this I get

(A

+ iB )( A + iB ) = A 2 + 2iAB + i 2 B 2 = A 2 − B 2 + 2iAB

(

)

This obviously has real part and an imaginary part. But the Born interpretation disallows such a square: a probability has to be real and positive. Negative or imaginary probabilities have no meaning! So when dealing with complex numbers we form the square by multiplying a number with its complex conjugate (you just reverse the sign of the imaginary part). In our example, the complex conjugate will be A – iB And then the square is simply

(A

− iB )( A + iB ) = A 2 + iAB − iAB − i 2 B 2 = A 2 + B 2

(6.18)

This will always be a real number: and it will always be positive too as the square of any real number like A or B will be positive. In general, if we represent a wavefunction by the function Ψ ( x ) , then its complex conjugate is given by Ψ ( x ) * and thus the square of a wavefunction is described as

Ψ 2 = Ψ(x ) Ψ(x ) ∗

(6.19)

Formally, we can state the Born Interpretation of the wavefunction as follows; the probability that a particle lies in the interval dx, located at the position x, is given by Probability = Ψ ( x ) Ψ ( x )dx ∗

(6.20)

Ψ( x ) Ψ (x ) is known as the probability density because you multiply by dx to give you the probability. It is ∗

always real and positive.

Please bare in mind that x is just a variable, it doesn’t have to be a distance or length. It could be an angle for example, which is the case for a hydrogen atom! Believe it or, we are at the point where we can completely explain both quantization of energy levels and the quantum numbers n, l and ml. It’s so exciting! This leads us nicely to today’s question: •

If we can set up the Schrödinger equation for hydrogen and solve it, then shouldn’t we be able to find those quantum numbers?

Key points in lecture 6 • •

Quantum objects such as electrons and photons are described by a mathematical function called the wavefunction. This often contains complex numbers. The wavefunction of any quantum object can be determined by solving the Schrödinger equation, which can also be used to calculate the energy of a particle.

Hˆ Ψ = EΨ •

• • •

Solving the Schrödinger equation for a free-body clearly demonstrates that wave-particle duality is necessary but not sufficient to create quantization of energy: something else is needed. The curvature of a wavefunction is a manifestation of the kinetic energy of a particle. The square of the wavefunction at a specific point is, according to the Born interpretation, proportional to the probability of measuring a particle at that point. Fascinating piece of useless information: Olivia Newton-John (Sandy in the film version of “Grease”) is Max Born’s granddaughter.