Lecture 6. Four postulates of quantum mechanics. The eigenvalue equation. Momentum and energy operators. Dirac delta function. Expectation values

Transcription

1 Lecture 6 Four postulates of quantum mechanics The eigenvalue equation Momentum and energy operators Dirac delta function Expectation values

2 Objectives Learn about eigenvalue equations and operators. Learn how wavefunctions change after making a measurement. Learn how to calculate an expectation value from a wavefunction and an operator.

3 Four postulates of quantum mechanics I For any observable, A, there is an operator A such that a measurement of A yields values,, which are eigenvalues of A. i.e. A = (eigenvalue equation) A can be something measurable like momentum, energy or position. A is an operator corresponding to something measurable. An operator is something that operates on a function to produce a new function. is an eigenfunction of A, we need to find values of which satisfy the eigenvalue equation. Example : If A=d / dx then =e nx is a possible solution to the eigenvalue equation as A =ne nx =n, so n is the eigenvalue of A.

4 What is an eigenvalue? Consider a vector in a 3D space with basis vectors (an orthogonal axis system) x, y and z. Suppose we transform the vector by rotating it by some angle about the x axis. Let s call the transformation operator T. If lies along the x axis then T = i.e. is unchanged. is an eigenvector of T. If lies in the y z plane and T is a 180 rotation about x then T =. Thus, for every transform T we can find vectors such that T =. In this example is an eigenvector, but the same argument holds for functions, in which case we replace with an eigenfunction,.

5 The momentum operator If the potential energy is zero then E= p 2 /2 m and the TISE reads p 2 ħ2 = 2 m 2 m d 2 ħ2 2 = dx 2 m 2. So p= i ħ is a solution. In 1D then p= i ħ x. The eigenvalue equation for p becomes i ħ x = p x, so if we make a measurement of the momentum we will get values p x. Suppose we have a free particle with = Aexp ip x x ħ = Aeikx k= p x / ħ, this satisfies the eigenvalue equation. The particle is represented by a periodic function (a wave) with e ikx =e ik x, i.e. 1=e ik and = 2 n. k For n=1 =2 / k and so =2 ħ/ p x =h/ p the de Broglie relation.

6 The energy operator The energy operator, H, is also known as the Hamiltonian. Classically, the energy of an object is the sum of the kinetic and potential energies. In quantum mechanics we have H = p2 V r, where the left 2 m term represents the kinetic energy and the right term the potential energy. We know that p= i ħ, so we can write H = ħ2 2 m 2 V r. So when we make a measurement of energy we are looking for solutions to the eigenvalue equation H =E. This is the same as the time independent Schrödinger equation.

7 Energy of a free particle A free particle has no potential energy so the Hamiltonian becomes H = ħ2 2 m 2. If the particle moves in one dimension the eigenvalue equation is ħ2 d 2 2 m dx =E. Let k= 2 m E and we have d 2 2 ħ dx = k 2. 2 A possible solution is = Ae ikx Be ikx, which has energy E= k 2 ħ 2 2 m. Note that = Ae ikx and =Be ikx are also eigenfunctions of H. Furthermore, = Ae ikx is also an eigenfunction of p, so a free particle with this eigenfunction has energy of E= k 2 ħ 2 2 m and momentum of p=ħ k. The energy and momentum are determined exactly, but the position cannot be determined as the wavefunction cannot be normalised, there is an equal probability of finding the particle at any location along the x axis.

8 Four postulates of quantum mechanics II Measurement of the observable A that yields the value leaves the system in the state, where is the eigenfunction of A that corresponds to the eigenvalue. If we measure the position, x, of a particle and obtain the result x= then we have x =. Before the measurement could be anything, after the measurement we have established a value for x and collapses to. We represent by the Dirac delta function = x. This function has an area of 1, but has infinitesimally narrow width about such that x =0 for x. It can be considered as a spike located at x=. If we have a function f x then f x x = f x, i.e. the Dirac delta function represents a simple multiplication by f.

9 The Dirac delta function The Dirac delta function is defined as follows: x =0 if x 0 and x dx=1 x =1 if x=0 It is an infinitely high, infinitesimally narrow spike located at the origin whose area is 1. The function x a is a spike at x=a. Consider these integrals f x x a dx= f a x a dx= f a The delta function is zero everywhere except where x=a. When multiplied by a function f x it has the same effect as multiplying by f a.

10 Effect on of making a measurement Wavefunction before and after measurement evolves with time according to the Schrödinger equation. If we make a measurement on the system we know (with some degree of error) where the particle is. This causes a collapse of the wavefunction.

11 Four postulates of quantum mechanics - III The state of a system at any instant of time may be represented by a state or wavefunction which is continuous and differentiable. All information regarding the state of the system is contained in the wavefunction. If a system is in the state x,t, the average of any observable A relevant to the system at time t is A = * A dx Where A is known as the expectation value of A.

12 Expectation values If we have a distribution of values, e.g. ages a, we calculate the a N a average age using a = a P a. N total = a=0 In quantum mechanics P a = x,t 2, so if we want to measure the position of a particle x is the average value (the expectation value) of x. i.e. x = x x,t 2 dx. What this means is that if we make measurements of x on an ensemble of identical systems x will be the average result. This is not the same as making many measurements of x on the same system as the state of the system will be changed after each measurement of x, may also be time dependent.

13 Expectation values An example Suppose x =Ae x x 0 2, what is x? First, normalise : A 2 e 2 x x 0 2 dx=1, put y=x x 0 then A 2 e 2y 2 dy=1. So A 2 2 =1, A= 2 1/4. The normalised wavefunction is = 2 1/4 e x x 0 2. The expectation value is x = x 2 dx or x = 2 1/2 x e 2 x x 0 2 dx. Again put y=x x 0 so x = 2 1/2 y x 0 e 2y2 dy. The term ye 2y 2 disappears on integration so the result is x = 2 1/2 x 0 2 1/2=x 0. Since the wavefunction is a Gaussian centred on x=x 0, this is what we would expect.

14 Time dependence of x The expectation value of momentum, p, is given by p =m d x dt let's calculate d x. dt, so d x dt = x t 2 dx & t 2 = i ħ 2 2 m * x 2 * 2 x = [ i ħ 2 x 2 m * ] * x x So d x dt = i ħ x * 2 m x * x x dx. Integrate by parts to get d x dt Integrating again gives d x dt = i ħ 2 m * = i ħ m * x dx. * x x dx.

15 We found that d x dt Expectation value of momentum p = v = i ħ m * x dx, the expectation value of velocity. So for momentum we have p =m v = i ħ * x dx. Another way of writing the expectation values is x = * x dx and p = * ħ i x dx The operator x represents position and the operator ħ i momentum. x represents More generally A = * A dx.

16 Hermitian operators In our discussion of expectation values we were able to write A = A * dx= * A dx We can only do this because operators which correspond to real observables are Hermitian. Proof for the momentum operator p= i ħ p = p * dx= i ħ * x dx, note the - sign has disappeared because we take the complex conjugate of p. Integrate by parts to get p =i ħ [ * ] i ħ * x dx= * p dx.

17 Four postulates of quantum mechanics IV The wavefunction x,t for a system develops in time according to the equation i ħ ħ2 2 = t 2 m x V or i ħ 2 t = H where H is known as the Hamiltonian. This is the same as the Schrödinger equation.

18 Conclusions The Schrödinger equation is an eigenvalue equation which means that measured physical properties such as momentum and position are eigenvalues, i.e. only certain values are allowed for a given system.