Page 1 Quantum Mechanics V H Satheeshkumar Department of Physics and Center for Advanced Research and Development Sri Bhagawan Mahaveer Jain College of Engineering Jain Global Campus, Kanakapura Road Bangalore 562 112, India. vhsatheeshkumar@gmail.com October 12, 2008 Who can use this? The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the second of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for the ?rst-semester (September 2008 - January2009) BE students of all branches. Any student interested in exploring more about the course may visit the course homepage at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-) Syllabus as prescribed by VTU Heisenberg’s uncertainty principle and its physical significance, Application of uncertainty principle; Wave function, Properties and Physical significance of a wave function, Probability density and Normalisation of wave function; Setting up of a one dimensional time independent, Schr¨ odinger wave equation, Eigen values and eigen function, Application of Schr¨ odinger wave equation : Energy eigen values for a free particle, Energy eigen values of a particle in a potential well of infinite depth. Reference • Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. ———————————— This document is typeset in Free Software L A T E X2e distributed under the terms of the GNU General Public License. 1 Page 2 Quantum Mechanics V H Satheeshkumar Department of Physics and Center for Advanced Research and Development Sri Bhagawan Mahaveer Jain College of Engineering Jain Global Campus, Kanakapura Road Bangalore 562 112, India. vhsatheeshkumar@gmail.com October 12, 2008 Who can use this? The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the second of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for the ?rst-semester (September 2008 - January2009) BE students of all branches. Any student interested in exploring more about the course may visit the course homepage at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-) Syllabus as prescribed by VTU Heisenberg’s uncertainty principle and its physical significance, Application of uncertainty principle; Wave function, Properties and Physical significance of a wave function, Probability density and Normalisation of wave function; Setting up of a one dimensional time independent, Schr¨ odinger wave equation, Eigen values and eigen function, Application of Schr¨ odinger wave equation : Energy eigen values for a free particle, Energy eigen values of a particle in a potential well of infinite depth. Reference • Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. ———————————— This document is typeset in Free Software L A T E X2e distributed under the terms of the GNU General Public License. 1 www.satheesh.bigbig.com/EnggPhy 2 1 Introduction Quantum mechanics is a fundamental branch of physics which generalizes classical mechanics to pro- vide accurate descriptions for many previously unexplained phenomena such as black body radiation, photoelectric e?ect and Compton e?ect. The term quantum mechanics was ?rst coined by Max Born in 1924. Within the ?eld of engineering, quantum mechanics plays an important role. The study of quantum mechanics has lead to many new inventions that include the laser, the diode, the transistor, the elec- tron microscope, and magnetic resonance imaging. Flash memory chips found in USB drives also use quantum ideas to erase their memory cells. The entire science of Nanotechnology is based on the quan- tum mechanics. Researchers are currently seeking robust methods of directly manipulating quantum states. E?orts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than the regular computers. This chapter attempts to give you an elementary introduction to the topic. 2 Heisenberg’s uncertainty principle We know from the wave-particle duality that every particle has wave-like properties. These wave prop- erties of particles will prevent us from measuring the exact attributes of the particles. This limitation related to the measurements at microscopic level is known as the uncertainty principle. The uncertainty principle states that it is impossible to specify simultaneously the position and momentum of a particle, such as an electron, with precision. The theory further states that a more accurate determination of one quantity will result in a less precise measurement of the other, and that the product of both uncertainties is always greater than or equal to Planck’s constant divided by 4p. That is ?x·?p x = h 4p . (1) This principle was formulated in 1927 by the German physicist Werner Heisenberg. It is also called the indeterminacy principle. The Heisenberg’s uncertainty principle can also be expressed in terms of the uncertainties involved in the simultaneous measurements of angular displacement & angular momentum and energy & time; ??·?l= h 4p , (2) ?t·?E = h 4p . (3) Sometimes h 2p is written as ~. In that case the right had side of the uncertainty relations will have ~ 2 . 2.1 Explanation of uncertainty principle using gamma ray microscope We use a hypothetical experiment of observing an electron using gamma ray microscope to illustrate the uncertainty principle. Suppose, we look at an electron using light of wavelength?. Each photon of this light has the momentum h/?. When one of these photons bounces o? the electron, the electron’s original momentum will be changed. The exact amount of the change ?p cannot be predicted, but it will be of the same order of magnitude as the photon momentum h/?. Hence ?p˜h/?. The longer the wavelength of the observing photon, the smaller the uncertainty in the electron’s mo- mentum. Page 3 Quantum Mechanics V H Satheeshkumar Department of Physics and Center for Advanced Research and Development Sri Bhagawan Mahaveer Jain College of Engineering Jain Global Campus, Kanakapura Road Bangalore 562 112, India. vhsatheeshkumar@gmail.com October 12, 2008 Who can use this? The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the second of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for the ?rst-semester (September 2008 - January2009) BE students of all branches. Any student interested in exploring more about the course may visit the course homepage at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-) Syllabus as prescribed by VTU Heisenberg’s uncertainty principle and its physical significance, Application of uncertainty principle; Wave function, Properties and Physical significance of a wave function, Probability density and Normalisation of wave function; Setting up of a one dimensional time independent, Schr¨ odinger wave equation, Eigen values and eigen function, Application of Schr¨ odinger wave equation : Energy eigen values for a free particle, Energy eigen values of a particle in a potential well of infinite depth. Reference • Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. ———————————— This document is typeset in Free Software L A T E X2e distributed under the terms of the GNU General Public License. 1 www.satheesh.bigbig.com/EnggPhy 2 1 Introduction Quantum mechanics is a fundamental branch of physics which generalizes classical mechanics to pro- vide accurate descriptions for many previously unexplained phenomena such as black body radiation, photoelectric e?ect and Compton e?ect. The term quantum mechanics was ?rst coined by Max Born in 1924. Within the ?eld of engineering, quantum mechanics plays an important role. The study of quantum mechanics has lead to many new inventions that include the laser, the diode, the transistor, the elec- tron microscope, and magnetic resonance imaging. Flash memory chips found in USB drives also use quantum ideas to erase their memory cells. The entire science of Nanotechnology is based on the quan- tum mechanics. Researchers are currently seeking robust methods of directly manipulating quantum states. E?orts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than the regular computers. This chapter attempts to give you an elementary introduction to the topic. 2 Heisenberg’s uncertainty principle We know from the wave-particle duality that every particle has wave-like properties. These wave prop- erties of particles will prevent us from measuring the exact attributes of the particles. This limitation related to the measurements at microscopic level is known as the uncertainty principle. The uncertainty principle states that it is impossible to specify simultaneously the position and momentum of a particle, such as an electron, with precision. The theory further states that a more accurate determination of one quantity will result in a less precise measurement of the other, and that the product of both uncertainties is always greater than or equal to Planck’s constant divided by 4p. That is ?x·?p x = h 4p . (1) This principle was formulated in 1927 by the German physicist Werner Heisenberg. It is also called the indeterminacy principle. The Heisenberg’s uncertainty principle can also be expressed in terms of the uncertainties involved in the simultaneous measurements of angular displacement & angular momentum and energy & time; ??·?l= h 4p , (2) ?t·?E = h 4p . (3) Sometimes h 2p is written as ~. In that case the right had side of the uncertainty relations will have ~ 2 . 2.1 Explanation of uncertainty principle using gamma ray microscope We use a hypothetical experiment of observing an electron using gamma ray microscope to illustrate the uncertainty principle. Suppose, we look at an electron using light of wavelength?. Each photon of this light has the momentum h/?. When one of these photons bounces o? the electron, the electron’s original momentum will be changed. The exact amount of the change ?p cannot be predicted, but it will be of the same order of magnitude as the photon momentum h/?. Hence ?p˜h/?. The longer the wavelength of the observing photon, the smaller the uncertainty in the electron’s mo- mentum. www.satheesh.bigbig.com/EnggPhy 3 Becauselightisawavephenomenonaswellasaparticlephenomenon,wecannotexpecttodetermine the electron’s location with perfect accuracy regardless of the instrument used. A reasonable estimate of the minimum uncertainty in the measurement might be one photon wavelength, so that ?x=?. The shorter the wavelength, the smaller the uncertainty in location. However, if we use light of short wavelength to increase the accuracy of the position measurement, there will be a corresponding decrease in the accuracy of the momentum measurement because the higher photon momentum will disturb the electron’s motion to a greater extent. Light of long wavelength will give a more accurate momentum but a less accurate position. Combining the above results gives us ?x·?p=?. This agrees well with the uncertainty principle. 2.2 Physical signi?cance of uncertainty principle The uncertainty principle is based on the assumption that a moving particle is associated with a wave packet, the extension of which in space accounts for the uncertainty in the position of the particle. The uncertainty in the momentum arises due to the indeterminacy of the wavelength because of the ?nite size of the wave packet. Thus, the uncertainty principle is not due to the limited accuracy of measurement but due to the inherent uncertainties in determining the quantities involved. Even though, the uncertainty principle prevents us from knowing the precise position and momentum, we can de?ne the position where the probability of ?nding the particle is maximum and also the most probable momentum of the particle. That means, the uncertainty principle introduces the probabilistic interpretation of the physical quantities. This is the major di?erence between the classical physics and quantum mechanics. 2.3 Application of uncertainty principle The uncertainty principle has far reaching implications. In fact, it has been very useful in explaining many observations which cannot be explained otherwise. An important one being the proof of the non-existence of an electron inside the nucleus. In beta decay, the electrons are emitted from the nucleus of the radioactive element. The radius of a typical atomic nucleus to be about 5.0×10 -15 m. Assuming that the uncertainty in the position of the electron inside the nucleus to be of the same order, we have ?x = 5.0×10 -15 m. The corresponding uncertainty in the momentum is, ?p x = h 4p · 1 ?x , ?p x = 6.63×10 -34 4×3.14 · 1 5.0×10 -15 , ?p x = 1.1×10 -20 kgms -1 . If this is the uncertainty in a nuclear electron’s momentum p itself must be at least comparable in magnitude. An electron with such a momentum has a kinetic energy,KE, many times greater than its rest energy (which is mc 2 ). The kinetic energy of such particle is given by KE =pc= (1.1×10 -20 )×(3×10 8 ) Page 4 Quantum Mechanics V H Satheeshkumar Department of Physics and Center for Advanced Research and Development Sri Bhagawan Mahaveer Jain College of Engineering Jain Global Campus, Kanakapura Road Bangalore 562 112, India. vhsatheeshkumar@gmail.com October 12, 2008 Who can use this? The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the second of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for the ?rst-semester (September 2008 - January2009) BE students of all branches. Any student interested in exploring more about the course may visit the course homepage at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-) Syllabus as prescribed by VTU Heisenberg’s uncertainty principle and its physical significance, Application of uncertainty principle; Wave function, Properties and Physical significance of a wave function, Probability density and Normalisation of wave function; Setting up of a one dimensional time independent, Schr¨ odinger wave equation, Eigen values and eigen function, Application of Schr¨ odinger wave equation : Energy eigen values for a free particle, Energy eigen values of a particle in a potential well of infinite depth. Reference • Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. ———————————— This document is typeset in Free Software L A T E X2e distributed under the terms of the GNU General Public License. 1 www.satheesh.bigbig.com/EnggPhy 2 1 Introduction Quantum mechanics is a fundamental branch of physics which generalizes classical mechanics to pro- vide accurate descriptions for many previously unexplained phenomena such as black body radiation, photoelectric e?ect and Compton e?ect. The term quantum mechanics was ?rst coined by Max Born in 1924. Within the ?eld of engineering, quantum mechanics plays an important role. The study of quantum mechanics has lead to many new inventions that include the laser, the diode, the transistor, the elec- tron microscope, and magnetic resonance imaging. Flash memory chips found in USB drives also use quantum ideas to erase their memory cells. The entire science of Nanotechnology is based on the quan- tum mechanics. Researchers are currently seeking robust methods of directly manipulating quantum states. E?orts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than the regular computers. This chapter attempts to give you an elementary introduction to the topic. 2 Heisenberg’s uncertainty principle We know from the wave-particle duality that every particle has wave-like properties. These wave prop- erties of particles will prevent us from measuring the exact attributes of the particles. This limitation related to the measurements at microscopic level is known as the uncertainty principle. The uncertainty principle states that it is impossible to specify simultaneously the position and momentum of a particle, such as an electron, with precision. The theory further states that a more accurate determination of one quantity will result in a less precise measurement of the other, and that the product of both uncertainties is always greater than or equal to Planck’s constant divided by 4p. That is ?x·?p x = h 4p . (1) This principle was formulated in 1927 by the German physicist Werner Heisenberg. It is also called the indeterminacy principle. The Heisenberg’s uncertainty principle can also be expressed in terms of the uncertainties involved in the simultaneous measurements of angular displacement & angular momentum and energy & time; ??·?l= h 4p , (2) ?t·?E = h 4p . (3) Sometimes h 2p is written as ~. In that case the right had side of the uncertainty relations will have ~ 2 . 2.1 Explanation of uncertainty principle using gamma ray microscope We use a hypothetical experiment of observing an electron using gamma ray microscope to illustrate the uncertainty principle. Suppose, we look at an electron using light of wavelength?. Each photon of this light has the momentum h/?. When one of these photons bounces o? the electron, the electron’s original momentum will be changed. The exact amount of the change ?p cannot be predicted, but it will be of the same order of magnitude as the photon momentum h/?. Hence ?p˜h/?. The longer the wavelength of the observing photon, the smaller the uncertainty in the electron’s mo- mentum. www.satheesh.bigbig.com/EnggPhy 3 Becauselightisawavephenomenonaswellasaparticlephenomenon,wecannotexpecttodetermine the electron’s location with perfect accuracy regardless of the instrument used. A reasonable estimate of the minimum uncertainty in the measurement might be one photon wavelength, so that ?x=?. The shorter the wavelength, the smaller the uncertainty in location. However, if we use light of short wavelength to increase the accuracy of the position measurement, there will be a corresponding decrease in the accuracy of the momentum measurement because the higher photon momentum will disturb the electron’s motion to a greater extent. Light of long wavelength will give a more accurate momentum but a less accurate position. Combining the above results gives us ?x·?p=?. This agrees well with the uncertainty principle. 2.2 Physical signi?cance of uncertainty principle The uncertainty principle is based on the assumption that a moving particle is associated with a wave packet, the extension of which in space accounts for the uncertainty in the position of the particle. The uncertainty in the momentum arises due to the indeterminacy of the wavelength because of the ?nite size of the wave packet. Thus, the uncertainty principle is not due to the limited accuracy of measurement but due to the inherent uncertainties in determining the quantities involved. Even though, the uncertainty principle prevents us from knowing the precise position and momentum, we can de?ne the position where the probability of ?nding the particle is maximum and also the most probable momentum of the particle. That means, the uncertainty principle introduces the probabilistic interpretation of the physical quantities. This is the major di?erence between the classical physics and quantum mechanics. 2.3 Application of uncertainty principle The uncertainty principle has far reaching implications. In fact, it has been very useful in explaining many observations which cannot be explained otherwise. An important one being the proof of the non-existence of an electron inside the nucleus. In beta decay, the electrons are emitted from the nucleus of the radioactive element. The radius of a typical atomic nucleus to be about 5.0×10 -15 m. Assuming that the uncertainty in the position of the electron inside the nucleus to be of the same order, we have ?x = 5.0×10 -15 m. The corresponding uncertainty in the momentum is, ?p x = h 4p · 1 ?x , ?p x = 6.63×10 -34 4×3.14 · 1 5.0×10 -15 , ?p x = 1.1×10 -20 kgms -1 . If this is the uncertainty in a nuclear electron’s momentum p itself must be at least comparable in magnitude. An electron with such a momentum has a kinetic energy,KE, many times greater than its rest energy (which is mc 2 ). The kinetic energy of such particle is given by KE =pc= (1.1×10 -20 )×(3×10 8 ) www.satheesh.bigbig.com/EnggPhy 4 KE = 3.3×10 -12 J KE = 20MeV This means that the kinetic energy of an electron must exceed 20MeV if it is to be inside a nucleus. Experiments show that the electrons emitted by certain unstable nuclei never have more than a small fraction of this energy, from which we conclude that nuclei cannot contain electrons. The electron that an unstable nucleus may emit comes into being only at the moment the nucleus decays. 3 Wave function In quantum mechanics, because of the wave-particle duality, the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape and extending over all of space. This is called a wave function. The wave function is usually complex and is represented by ?. Since the wave function is complex, its direct measurement in any physical experiment is not possible. It is just mathematical function of x, t etc. Once the wave function corresponding to a system is known, the state of the system can be determined. The physical state of system is completely characterized by a wave function. 3.1 Physical signi?cance of a wave function The wave function contains informationabout the system itrepresents. Even though the wave function itself is not directly an observable quantity, the square of the absolute value of the wave function gives the probability of ?nding the particle at a given space and time. This probabilistic interpretation of wave function was given by Max Born in 1926. If ? is the wave function associated with a particle, the|?| 2 is the probability per unit volume that the particle will be found at the given point. The probability density is given by |?| 2 = ?·? * where ? * is the complex conjugate of ?. For a particle restricted to move only long x- axis, the probability of ?nding it between x 1 and x 2 is given by Z x 2 x 1 |?| 2 dx. Since the probability of ?nding a particle any where in a given voluve must be one, we have Z +8 -8 |?| 2 dV = 1. This condition is know as normalization. 3.2 Properties of a wave function A wave function has the following characteristics. 1. ? must be continuous and single-valued everywhere. 2. ??/?x, ??/?y and ??/?z must be continuous and single-valued everywhere. 3. ? must be normalizable. Page 5 Quantum Mechanics V H Satheeshkumar Department of Physics and Center for Advanced Research and Development Sri Bhagawan Mahaveer Jain College of Engineering Jain Global Campus, Kanakapura Road Bangalore 562 112, India. vhsatheeshkumar@gmail.com October 12, 2008 Who can use this? The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the second of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for the ?rst-semester (September 2008 - January2009) BE students of all branches. Any student interested in exploring more about the course may visit the course homepage at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-) Syllabus as prescribed by VTU Heisenberg’s uncertainty principle and its physical significance, Application of uncertainty principle; Wave function, Properties and Physical significance of a wave function, Probability density and Normalisation of wave function; Setting up of a one dimensional time independent, Schr¨ odinger wave equation, Eigen values and eigen function, Application of Schr¨ odinger wave equation : Energy eigen values for a free particle, Energy eigen values of a particle in a potential well of infinite depth. Reference • Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. ———————————— This document is typeset in Free Software L A T E X2e distributed under the terms of the GNU General Public License. 1 www.satheesh.bigbig.com/EnggPhy 2 1 Introduction Quantum mechanics is a fundamental branch of physics which generalizes classical mechanics to pro- vide accurate descriptions for many previously unexplained phenomena such as black body radiation, photoelectric e?ect and Compton e?ect. The term quantum mechanics was ?rst coined by Max Born in 1924. Within the ?eld of engineering, quantum mechanics plays an important role. The study of quantum mechanics has lead to many new inventions that include the laser, the diode, the transistor, the elec- tron microscope, and magnetic resonance imaging. Flash memory chips found in USB drives also use quantum ideas to erase their memory cells. The entire science of Nanotechnology is based on the quan- tum mechanics. Researchers are currently seeking robust methods of directly manipulating quantum states. E?orts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than the regular computers. This chapter attempts to give you an elementary introduction to the topic. 2 Heisenberg’s uncertainty principle We know from the wave-particle duality that every particle has wave-like properties. These wave prop- erties of particles will prevent us from measuring the exact attributes of the particles. This limitation related to the measurements at microscopic level is known as the uncertainty principle. The uncertainty principle states that it is impossible to specify simultaneously the position and momentum of a particle, such as an electron, with precision. The theory further states that a more accurate determination of one quantity will result in a less precise measurement of the other, and that the product of both uncertainties is always greater than or equal to Planck’s constant divided by 4p. That is ?x·?p x = h 4p . (1) This principle was formulated in 1927 by the German physicist Werner Heisenberg. It is also called the indeterminacy principle. The Heisenberg’s uncertainty principle can also be expressed in terms of the uncertainties involved in the simultaneous measurements of angular displacement & angular momentum and energy & time; ??·?l= h 4p , (2) ?t·?E = h 4p . (3) Sometimes h 2p is written as ~. In that case the right had side of the uncertainty relations will have ~ 2 . 2.1 Explanation of uncertainty principle using gamma ray microscope We use a hypothetical experiment of observing an electron using gamma ray microscope to illustrate the uncertainty principle. Suppose, we look at an electron using light of wavelength?. Each photon of this light has the momentum h/?. When one of these photons bounces o? the electron, the electron’s original momentum will be changed. The exact amount of the change ?p cannot be predicted, but it will be of the same order of magnitude as the photon momentum h/?. Hence ?p˜h/?. The longer the wavelength of the observing photon, the smaller the uncertainty in the electron’s mo- mentum. www.satheesh.bigbig.com/EnggPhy 3 Becauselightisawavephenomenonaswellasaparticlephenomenon,wecannotexpecttodetermine the electron’s location with perfect accuracy regardless of the instrument used. A reasonable estimate of the minimum uncertainty in the measurement might be one photon wavelength, so that ?x=?. The shorter the wavelength, the smaller the uncertainty in location. However, if we use light of short wavelength to increase the accuracy of the position measurement, there will be a corresponding decrease in the accuracy of the momentum measurement because the higher photon momentum will disturb the electron’s motion to a greater extent. Light of long wavelength will give a more accurate momentum but a less accurate position. Combining the above results gives us ?x·?p=?. This agrees well with the uncertainty principle. 2.2 Physical signi?cance of uncertainty principle The uncertainty principle is based on the assumption that a moving particle is associated with a wave packet, the extension of which in space accounts for the uncertainty in the position of the particle. The uncertainty in the momentum arises due to the indeterminacy of the wavelength because of the ?nite size of the wave packet. Thus, the uncertainty principle is not due to the limited accuracy of measurement but due to the inherent uncertainties in determining the quantities involved. Even though, the uncertainty principle prevents us from knowing the precise position and momentum, we can de?ne the position where the probability of ?nding the particle is maximum and also the most probable momentum of the particle. That means, the uncertainty principle introduces the probabilistic interpretation of the physical quantities. This is the major di?erence between the classical physics and quantum mechanics. 2.3 Application of uncertainty principle The uncertainty principle has far reaching implications. In fact, it has been very useful in explaining many observations which cannot be explained otherwise. An important one being the proof of the non-existence of an electron inside the nucleus. In beta decay, the electrons are emitted from the nucleus of the radioactive element. The radius of a typical atomic nucleus to be about 5.0×10 -15 m. Assuming that the uncertainty in the position of the electron inside the nucleus to be of the same order, we have ?x = 5.0×10 -15 m. The corresponding uncertainty in the momentum is, ?p x = h 4p · 1 ?x , ?p x = 6.63×10 -34 4×3.14 · 1 5.0×10 -15 , ?p x = 1.1×10 -20 kgms -1 . If this is the uncertainty in a nuclear electron’s momentum p itself must be at least comparable in magnitude. An electron with such a momentum has a kinetic energy,KE, many times greater than its rest energy (which is mc 2 ). The kinetic energy of such particle is given by KE =pc= (1.1×10 -20 )×(3×10 8 ) www.satheesh.bigbig.com/EnggPhy 4 KE = 3.3×10 -12 J KE = 20MeV This means that the kinetic energy of an electron must exceed 20MeV if it is to be inside a nucleus. Experiments show that the electrons emitted by certain unstable nuclei never have more than a small fraction of this energy, from which we conclude that nuclei cannot contain electrons. The electron that an unstable nucleus may emit comes into being only at the moment the nucleus decays. 3 Wave function In quantum mechanics, because of the wave-particle duality, the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape and extending over all of space. This is called a wave function. The wave function is usually complex and is represented by ?. Since the wave function is complex, its direct measurement in any physical experiment is not possible. It is just mathematical function of x, t etc. Once the wave function corresponding to a system is known, the state of the system can be determined. The physical state of system is completely characterized by a wave function. 3.1 Physical signi?cance of a wave function The wave function contains informationabout the system itrepresents. Even though the wave function itself is not directly an observable quantity, the square of the absolute value of the wave function gives the probability of ?nding the particle at a given space and time. This probabilistic interpretation of wave function was given by Max Born in 1926. If ? is the wave function associated with a particle, the|?| 2 is the probability per unit volume that the particle will be found at the given point. The probability density is given by |?| 2 = ?·? * where ? * is the complex conjugate of ?. For a particle restricted to move only long x- axis, the probability of ?nding it between x 1 and x 2 is given by Z x 2 x 1 |?| 2 dx. Since the probability of ?nding a particle any where in a given voluve must be one, we have Z +8 -8 |?| 2 dV = 1. This condition is know as normalization. 3.2 Properties of a wave function A wave function has the following characteristics. 1. ? must be continuous and single-valued everywhere. 2. ??/?x, ??/?y and ??/?z must be continuous and single-valued everywhere. 3. ? must be normalizable. www.satheesh.bigbig.com/EnggPhy 5 4 Time independent Schr¨ odinger wave equation in one di- mension In quantum mechanics, the Schrdinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton’s laws are to classical mechanics. The equation is named after Erwin Schrdinger, who discovered it in 1926. Consider a wave function of an arbitrary particle ?(x,t) =Ae -i(?t-kx) . (4) Using the de?nitions of ? and k, we write the following ? = 2p? and k = 2p ? . From Planck’s law we have E =h? and substituting in the ? equation ? = 2p E h = E h/2p = E ~ . From de Bbroglie’s equation, we have ? =h/p and substituting in the k equation k = 2p h/p = p h/2p = p ~ . Now, we substitute the new expressions for? andk in the equation of the wave function. This gives us ?(x,t) =Ae -i ~ (Et-px) . We re-write the wave function with separate space and time parts ?(x,t) =Ae -iEt ~ ·e ipx ~ , ?(x,t) =fe -iEt ~ , (5) where f =Ae ipx ~ . (6) Di?erentiating the function f with respect to x twice, ?f ?x = ip ~ ·Ae ipx ~ ? 2 f ?x 2 = ip ~ · ip ~ ·Ae ipx ~ , that is ? 2 ? ?x 2 = -p 2 ~ 2 f, From here, we can write p 2 f =-~ 2 ? 2 f ?x 2 (7)Read More

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