3 Wave Particle Duality in Young’s double slit Experiment & its Quantum Mechanical interpretation
Prof. V.S. Bhasin
TABLE OF CONTENTS
1. Learning Outcomes
2. Introduction
3. Wave-Particle Duality and its Quantum Mechanical Interpretation 3.1 Young’s double slit Experiment
3.1.1 Quantum Mechanical Viewpoint
3.2 Introduction of Wave function and Probability Distribution
3.2.1 Statistical Interpretation of the Wave Function
3.2.2 Normalization of
4. The Principle of Superposition
5. Wave Functions for Particles having a definite momentum and energy 5.1 Energy and Momentum acting
as differential operators
6. Summary
1. Learning Outcomes
- After studying this module, you shall be able to
- Study the duality aspect of photons and quantum particles through Young’s double slit experiment
- Learn that it is an inherent property of photon to behave simultaneously as a wave and particle…
- Know that the classical picture that each particle or a photon has a definite and deterministic position at any instant has to be discarded
- Learn that wave nature enables one to calculate only the probability of a manifestation of photon
- Know the description of a quantum mechanical particle by introducing a wave function in terms of which probability distribution is defined.
- Learn the superposition principle of wave functions
- Study the wave functions of particles having definite energy and momentum
- Learn how energy and momentum act as operators operating on the wave function
2. Introduction
The discussion of several phenomena along with the corresponding experimental data studied in the preceding two modules clearly showed conclusively that both electromagnetic radiation and material particles at microscopic level possess dual nature, i.e., radiation behaving as wave can also exhibit the discrete photon character and similarly discrete nature of material atomic particle behaving as a wave. In this module, we analyze this aspect of duality by revisiting Young’s double slit experiment. We shall arrive at an important conclusion that light behaves simultaneously as a wave and as a stream of independent photons . The wave nature, which appears to be an inherent property of photon, enables us to calculate the probability of the manifestation of a photon. The interference pattern is a statistical property of photon. Thus the classical picture that each particle or a photon has a definite and deterministic position at any instant has to be discarded. From a stand point of quantum mechanics, the interference pattern would be destroyed, if a sufficiently careful attempt is made to determine through which slit each photon was passed. We generalize this picture to introduce a wave function for a quantum mechanical particle in terms of which the probability distribution is defined. The principle of superposition for two wave functions is also discussed. The wave function describing the motion of a free particle with definite momentum and energy is introduced and it is shown how these quantities act as operators on the wave function.
3. Wave-Particle Duality— Quantum Mechanical Interpretation 3.1 Young’s Double Slit Experiment
Let us consider the well-known Young’s double experiment from the point of view of analyzing the dual aspect of radiation. Suppose we have a mono-energetic beam of photons emitted by a source and allow it to pass through the two slits 1 2 before they are incident on the screen D. We know from our earlier studies that interference fringes are formed on the screen due to superposition of the electromagnetic waves emitted from the two slits. The intensity of light at a point on the screen D is proportional to the square of the amplitude of the electric field at this point. If the two slits, 1 2 produce respectively the electric fields, 1(x) and 2 (x) , at a point x on the screen, the resultant electric field at that point is
where 1(x) and 2 (x) , being complex, the right hand side in Eqn.(3.2) represents modulus squared. The important point to note from this equation is that the intensity I(x) differs from ( ) and ( ) by an interference term which depends on the phase difference between 1(x) and 2 (x) . It is the interference term which accounts for the formation of fringes on the screen. The wave theory also predicts that by reducing the
Fig. 3.1 A schematic sketch of Young’s double slit experiment
the intensity of the source, the intensity of the fringes gets reduced but fringes do not disappear.
In terms of the photon picture, can we explain the formation of fringes by assuming that photons emitted from one slit interact with those emitted from the second slit? To answer this question, suppose we reduce the intensity of the source S (i.e., the number of photons emitted per second) until the photons strike the screen practically one by one thereby reducing the possibility of interaction between the photons and making it eventually vanish altogether. In such a scenario, we should expect that the interference fringes should vanish. But what we observe is that by covering the screen with a photographic plate and increasing the exposure time so as to collect a large number of photons on the plate, we would observe that the fringes have not disappeared. Thus the possibility of explaining the occurrence of interference fringes arising due to interaction between photons, i.e., on the basis of pure particle picture of light, can just be ruled out.
By exposing the photographic plate for a short duration, so short that it receives only a few photons, we would observe that each photon has only a localized effect and would not result in producing an interference ( howsoever weak) pattern. In actual practice, it is only when a large number of photons have reached the screen, does the distribution of their impacts starts appearing. The density of impacts at each point on the screen results in the intensity distribution corresponding to the interference pattern.
Since in the experiment, we have here excluded the possibility of photon-photon interactions , let us we make an attempt to determine through which slit each photon passed through before reaching the screen. For this purpose, suppose we place detectors behind the slits S1 and S2 . We will the find that if the photons arrive one by one, each one that passes through a particular slit will be recorded through a signal either by the detector placed behind 1 ℎ ℎ 2 . Clearly the photons detected in this way are being absorbed by the respective detectors and do not reach the screen. Let us, for instance, remove the detector which blocks only 1. Then the one which is behind 2 would tell us that out of a large number of photons, about half pass through 2. The others which reach the screen passing through 1do not develop the interference pattern, but produce, on the other hand, only a diffraction pattern.
Thus analyzing this experiment from the particle picture of photons, as possibility of photon-photon interactions are ruled out, each photon must be considered separately. The basic question which remains to be answered is this: why the phenomenon should change drastically depending on whether only one or both the slits are open? In other words, for a photon passing through one of the slits why should the fact that the other is open or closed make such a fundamental difference?
3.1.1 Quantum mechanical view point:
In the question just raised, we make the implicit assumption that the photon actually goes through a particular one of the two slits. From the point of view of classical theory, this assumption is natural since classically each particle or a photon has definite and deterministic position at each instant of time. At the quantum mechanical or a microscopic level, this picture needs to be discarded. For example, an essential characteristic of this new domain appeared when we placed detectors behind the slits by trying to detect through which slit the photons crossed the slits. In fact, a detailed analysis shows that it is impossible to observe the interference pattern and to know at the same time through which slit each photon had passed.
From the above experiment we have also seen that as the photons reach the screen one by one, their impacts gradually build up the interference pattern ; that is to say, for a particular photon, we can not anticipate where it will strike the screen. We can only say that the probability of its striking the screen at a point x is proportional to Here too, we discard another classical idea, according to which., ‘the motion of a particle at any time can be uniquely predicted from its motion at an earlier time’.
We can thus summarize the analysis of Young’s double slit experiment based on the concept of wave-particle duality, as:
(a) Light behaves simultaneously as a wave and as a stream of independent photons . The wave nature, which
appears to be an inherent property of photon, enables us to calculate the probability of the manifestation of
a photon.
(b) The probability amplitude of a photon appearing at a time t at the point r can be described by the
electromagnetic wave represented by E(r, t) and the corresponding probability is proportional to
3.2 Introduction of Wave function and Probability Distribution
Generalizing the conclusions arrived at from the above experiment to describe a quantum mechanical particle, we introduce, by analogy, that in quantum mechanics a wave function or state function ) , (x ,y ,z , t) which as we shall see is complex in nature, plays the role of a probability amplitude. Thus the probability
( x, y ,z , t ) of finding the particle at a particular point with coordinates x, y, z at a time t within a volume V is proportional to | | i.e.,
Note that probabilities are real numbers, though may be complex.
3.2.1 Statistical Interpretation of the Wave Function:
is defined the position probability density. This interpretation, given by Born, is basically a statistical one.
3.2.2 Normalization of :
Since the probability of finding the particle somewhere in the region must be unity, the wave function ( ⃗ , ) must be normalized to unity, i.e.,
where the integral is taken over all space. A wave function for which the integral over the space is finite is said to be square integrable. Note that since | ( ⃗ , )| is a physically measurable quantity, two wave functions which differ each other by a constant multiplicative factor of modulus one ( such as ,exp(iα) ), )are equivalent and they satisfy the same normalization condition.
4. The Principle of Superposition:
As we have seen in the case of Young’s double slit experiment, in order to account for interference effects, it must be possible to superpose the wave functions. Thus if one possible stste of an ensemble of identical systems is described by a wave function and another state of this ensemble by a wave function then any linear combination
Notice that unlike the classical waves such as sound waves, the wave function ( ⃗ , ) is an abstract quantity, which has a statistical interpretation. This provides a complete description of the dynamical state of the system.
5. Wave Functions for Particles having a definite momentum and energy
Let us consider a simple case of free particles to investigate how wave functions can be used to describe their behavior. We have already seen that with electromagnetic waves one associates a particle, i.e., a photon whose energy and magnitude of momentum are related to the frequency ν and wavelength λ of the electromagnetic radiation by
where we introduce the angular frequency ω=2лν and the wave number k=2л/λ and the reduced Planck’s constant ℏ = /2л .
Suppose we consider a free particle of mass m moving along x-axis with momentum,⃗ = ̂, where ̂ is a unit vector along the x-axis with energy E. We associate with this particle a wave travelling along the positive x-axis with a definite momentum or wave number k. We know such a wave is a plane wave represented by
where A is a constant. This wave has wavelength λ=2л/k and angular frequency ω. Using the relations given in Eq.(3.11), we can rewrite this expression as
5.1 Energy and Momentum acting as differential operators
By differentiating the above equation first with respect to x and again with respect to t, it can be easily checked that the wave function (3.13) satisfies the following relations:
The significance of these expressions will be clear shortly. In the above, we have considered the propagation of wave in one dimension, which can be easily generalized to three dimensions. Thus, a particle having well defined momentum ⃗ and an energy E has associated with it a plane wave given by
where the propagation vector ⃗ is related to the momentum ⃗ by ⃗ = ℏ ⃗ with k=⌈ ⃗⌉ = / . Thus, in the case of three dimensions, Eq.(3.14) can be generalized to
where ⃗ is the gradient operator. Eq.(3.15) remains unchanged for the 3-dimensional plane wave. The relations, Eqs.(3.17) and (3.15), show that for a free particle, the energy and momentum can be represented by the differential operators
which act on the wave function . This is a postulate of wave mechanics: the dynamical variables, E and ⃗ , are represented by the differential operators even when the particle is not free.
In the earlier section, we found that the probability interpretation of the wave functions requires that they be normalized to unity. For one dimensional systems, the normalization condition requires that
However, the plane wave given by Eq.(3.13) does not satisfy this requirement . It becomes clear, by substituting Eq.(3.13) in the above integral, that we shall have
Obviously, the integral does not exist. The reason is that a plane wave having a well defined momentum is spread over all the space and therefore has the same probability of finding the particle at any point in space. In other words, it is completely delocalized in space. Such wave functions require special considerations and will be discussed later. For the present, however, if a particle is to be localized in space, we shall be required to superpose plane waves corresponding to different momenta to form what is called a localized wave packet.
6. Summary:
- After studying this module, you shall be able to
- Study the duality aspect of photons and quantum particles through Young’s double slit experiment
- Learn that it is an inherent property of photon to behave simultaneously as a wave and particle…
- Know that the classical picture that each particle or a photon has a definite and deterministic position at any instant has to be discarded
- Learn that wave nature enables one to calculate only the probability of a manifestation of photon
- Know the description of a quantum mechanical particle by introducing a wave function in terms of which probability distribution is defined.
- Learn the superposition principle of wave functions
- Study the wave functions of particles having definite energy and momentum
- Learn how energy and momentum act as operators operating on the wave function
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References
For More Details ( on this topic and other topics discussed in Text Module) See
1.Quantum Mechanics by Bransden and Joachain, Pearson Education Ltd(2000).
2.Quantum Mechanics ,Vol. I, by C. Cohen- Tanoudji , John Wiley & Sons
3.A Text Book of Quantum Mechanics by P M Mathews and K Venkatesan; Tata McG raw-Hill Publishing Co.
Ltd.
4. L.D. Landau & E.M.Lifshitz (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. 3 (3rd ed.). Pergman
Press
5. Messiah, Quantum Mechanics, Vol. 1, Wiley, NY, 1961
For General Study on Origins of Quantum Theory:
1 Introducing Quantum Theory by J P Mc Evoy, University of Pittsburgh
2. The Origins of the Quantum Theory by Cathryn Carson