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Max Planck (18581947) Louis De Broglie (1892-1987)

Established equation E= hAttributed wave-like properties to particles

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The postulates of quantum mechanics

2.1 The five postulates of quantum mechanics

The formulation of quantum mechanics, also called wave mechanics focuses on the wavefunction, ( x, y, z, t

), which depends on the spatial coordinates x, y, z, and the time t. In thefollowing sections we shall restrict ourselves to one spatial dimension x, so that the wavefunction depends solely on x. An extension to three spatial dimensions can be done easily. Thewave function ( x, t) and its complex conjugate (x, t) are the focal point of quantum*

mechanics, because they provide a concrete meaning in the macroscopic physical world: Theproduct (x, t)( x, t) dxis the probability to find a particle, for example an electron, within the*

interval xand x + dx. The particle is described quantum mechanically by the wave function( x, t). The product (x, t)( x, t) is therefore called the window of quantum mechanics to the*

real world.Quantum mechanics further differs from classical mechanics by the employment of

operators rather than the use of dynamical variables . Dynamical variables are used in classicalmechanics, and they are variables such as position, momentum, or energy. Dynamical variablesare contrasted with static variables such as the mass of a particle. Static variables do not change

during typical physical processes considered here. In quantum mechanics, dynamical variablesare replaced by operators which act on the wave function. Mathematical operators aremathematical expressions that act on an operand. For example, (d / d x) is the differentialoperator. In the expression (d / d x ) ( x, t

), the differential operator acts on the wave function,( x, t), which is the operand. Such operands will be used to deduce the quantum mechanicalwave equation or Schrdinger equation.

The postulates of quantum mechanics cannot be proven or deduced. The postulates arehypotheses, and, if no violation with nature (experiments) is found, they are called axioms , i. e.non-provable, true statements.

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Postulate 1The wave function ( x, y, z, t

) describes the temporal and spatial evolution of a quantum-mechanical particle. The wave function ( x, t) describes a particle with one degree of freedomof motion.

Postulate 2

The product * (x, t) ( x, t) is the probability density function of a quantum-mechanical particle.*( (x, t

) x, t) dxis the probability to find the particle in the interval between xand x+ d x.Therefore,

8* (2.1)=(x , t) (x , t) dx 1

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If a wave function ( x, t) fulfills Eq. (2.1), then ( x, t) is called a normalizedwave function.Equation (2.1) is the normalization condition and implies the fact that the particle must belocated somewhere on the xaxis.

Postulate 3

The wave function ( x, t) and its derivative ( / x) ( x, t) are continuous in an isotropicmedium.

lim (x, t) = (x , t) (2.2)0

x x0

. (2.3)lim (x , t) = (x, t)x xx x x = x0 0

In other words, ( x, t

) is a continuous and continuously differentiable function throughoutisotropic media. Furthermore, the wave function has to be finite and single valued throughout

position space (for the one-dimensional case, this applies to all values of x).

Postulate 4

Operators are substituted for dynamical variables. The operators act an the wave function ( x , t).In classical mechanics, variables such as the position, momentum, or energy are calleddynamical variables. In quantum mechanics operators rather than dynamical variables areemployed. Table 2.1 shows common dynamical variables and their corresponding quantum-mechanical operators

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Dynamical variable in Quantum-mechanical

classical mechanics operator

x x (2.4)

f(x) f(x) (2.5)

h (2.6)p

i x

f(p) (2.7)f hi x

h (2.8)E - itotal

t

Table 2.1 : Dynamical variables and their corresponding quantum-mechanical operators.

We next substitute quantum mechanical operators for dynamical variables in the total energyequation (see Eq. 1.2.6)

2

p. (2.9)+ U(x ) = Etotal2 m

Using the substitutions of Eqs. (2.4) to (2.8), and inserting the operand ( x, t), one obtains theSchrdinger or quantum mechanical wave equation

2 2h h- + = -(x, t) U(x) (x, t) (x, t) . (2.10)

2 m 2 i t

x

The Schrdinger equation is, mathematically speaking, a linear, second order, partial differentialequation.

Postulate 5

The expectation value, , of any dynamical variable , is calculated from the wave function

according to

8 *= (x, t) (x, t) dx (2.11)op- 8

where is the operator of the dynamical variable . The expectation value of a variable is alsoop

referred to as average value or ensemble average, and is denoted by the brackets . Equation...

(2.11) allows one to calculate expectation values of important quantities, such as the expectationvalues for position, momentum, potential energy, kinetic energy, etc.

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The five postulates are a concise summary of the principles or quantum mechanics. Thepostulates have severe implications on the interpretation of microscopicphysical processes. Onthe other hand, quantum-mechanics smoothly merges into newtonian mechanics for macroscopic

physical processes.The wave function ( x, t

) depends on time. As will be seen in the Section on Schrdingersequation, the time dependence of the wave function can be separated from the spatialdependence. The wave function can then be written as

i t( (2.12)

x , t) = (x) e

where ( x) is stationary and it depends only on the spatial coordinate. The harmonic timedependence of ( x , t

) is expressed by the exponential factor exp (i t).

An example of a stationary wave function is shown in Fig. 2.1 and this wave function is usedto illustrate some of the implications of the five postulates. It is assumed that a particle isdescribed by the wave function

) for

(x) = A (1+ cosx x< p (2.13)

0 ( =

x) for x= p . (2.14)

According to the second postulate, the wave function must be normalized, i. e.

8* ( =

x) (x) dx 1. (2.15)- 8

This condition yields the constant and thus the normalized wave function is givenA = 3

1 / p

by

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1for x< p (2.16)(x ) = (1 + cos x)

3 p

0 ( =

x) for x= p . (2.17)

Note that ( x) is a continuous function and is continuously differentiable throughout positionspace.

The potential energy of the particle, whose wave function is given by Eqs. (2.16) and (2.17),has a minimum probably around x= 0. A guess of such a potential is shown in the lower part ofFig. 2.1. A particle in such a potential experiences a force towards the potential minimum (seeEq. 2.4). Therefore, the corresponding wave function will be localized around the potential-minimum.

Next some expectation values associated with wave function shown in Fig. 2.1 will becalculated using the fifth Postulate. The position expectation value of a particle described by thewave function ( x) is given by

8* . (2.18)=x (x) x (x ) dx

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Note that xis now an operator, which acts on the wave function ( x). Note further that x( x) isan odd function, and since *( x) is even, the integrand *( x) x( x) is again an odd function.The integral over an odd function is zero, i. e.

x = 0 . (2.19)

Thus, the expectation value of the position is zero. In other words, the probability to find theparticle at any given time is highest at x= 0.

It is interesting to know, how far the wave function is distributed from its expectation value.In statistical mathematics, the standard deviation of any quantity, e. g. , is defined as

22 1 / 2 (2.20)- .

A measure of the spatial extent of the wave function is the standard deviation of the position ofthe particle. Hence, the spatial standard deviation of the particle on the xaxis is given by

1/ 2- 22s =

x x . (2.21)

With = 0 one obtains

p 2p 52 * 2 . (2.22)= = -x (x ) x (x) dx3 2

- p

> - < > > )The standard deviation s = ( < x

x ) = (< x is shown in Fig. 2.1 and it is a measure of2 2 1/2

2 1/2

the spatial extent of the wave function.

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The expectation value of the particle momentum can be determined in an analogous way

8h* . (2.23)p = (x ) (x) dxi x

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Evaluation of the integral yields < p> = 0 . In other words, the particle has no net momentum andit remains spatially at the same location, which is evident for a stationary wave function.

Similarly, the expectation values of kinetic energy, potential energy, and total energy can becalculated if ( x) and U(x) are known. The expectation values of these quantities are given by:

8 2 2-

h (2.24)*=Kinetic energy: E (x) (x) dxkin 2 m 2

x- 8

8*Potential energy: (2.25)=U (x) U(x ) (x ) dx

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Total energy:8 + 2 2- h* (2.26)= + =E E U (x) U(x) (x) dxtotal kin 2 m 2

x- 8

2.2 The de Broglie hypothesis

The de Broglie hypothesis (de Broglie, 1923) is a significant milestone in the development ofquantum mechanics because the dualism of waves and matter finds its synthesis in thishypothesis. Typical physical properties that had been associated with matter before the advent ofquantum mechanics are mass , velocity , and momentum . On the other hand, wavelength , phase-velocity, and group-velocityhad been associated with waves. The bridge between the world ofwaves and the corpuscular world is the de Broglie relation

(2.27)

= h / p

which attributes a vacuum wavelength to a particle with momentum p. This relation, which deBroglie postulated in 1923, can also be written as

= (2.28)

hp k

where k= 2 p / is the wavenumber. The kinetic energy of a classical particle can then beexpressed in terms of its wavenumber, that is

2 2 2p h k. (2.29)= =Ekin 2 m 2 m

Four years after de Broglies hypothesis, Davisson and Germer (1927) demonstratedexperimentally that a wavelength can be attributed to an electron, i. e. a classical particle. They

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found, that a beam of electrons with momentum p and wavelength was diffracted by a Ni-crystalthe same way as x-rays of the same wavelength . The relation between electron momentum pand the x-ray wavelength , which yields the same diffraction pattern, is given by the de Broglieequation, Eq. (2.27). Thus, a bridge between particles and waves had been built. No longer couldone think of electrons as pure particles or x-rays as pure waves. The nature of small particles has

both, particle-like and wave-like characteristics. Analogously, a wave has both, wave-like andparticle-like characteristics. This fact is known as the dual nature of particles and waves.

A simple diffraction experiment is shown in Fig. 2.2. A beam of particles incidents on ascreen with a slit. A diffraction pattern is detected on a screen behind the slit as shown in thelower part. Electrons and x-rays with the same energy generate the same diffraction pattern. Thediffraction pattern can be calculated by taking into account the constructive and destructiveinterference of waves.

The Davisson and Germer experiment further shows that the deterministic nature of classicalmechanics is not valid for quantum mechanical particles. No longer is it possible to predict orcalculate the exact trajectory of a particle. Instead, one can only calculate probabilities(expectation values). For example, the position expectation value of an electron passing through

> = 0.the slit of Fig. 2.2 is