AP - Unit 1 College Notes PDF

Module-I QUANTUM MECHANICS Introduction: Quantum mechanics is a new branch of study in physics, which is indispensable in understanding the mechanics of particles in the atomic and sub-atomic scale. The motion of macro particles can be observed either direc...

Module-I QUANTUM MECHANICS Introduction: Quantum mechanics is a new branch of study in physics, which is indispensable in understanding the mechanics of particles in the atomic and sub-atomic scale. The motion of macro particles can be observed either directly or through microscope. Classical mechanics can be applied to explain the mechanics of macro particles. However, classical mechanics failed to explain the motion of micro particles like electrons, protons etc. Failures of Classical Mechanics : 1. Black Body radiation 2. Specific Heat of solids at low temperature 3. Theory of atomic structure 4. Photo – electric effect and 5. Compton effect. Max Plank proposed the Quantum Theory to explain Blackbody radiation. Einstein applied it to explain the Photo Electric Effect. In the mean time, Einstein’s mass – energy relationship (E = mc2) had been verified in which the radiation and mass were mutually convertible. Louis de-Broglie extended the idea of dual nature of radiation to matter, when he proposed that matter possess wave as well as particle characteristics. Assumptions of Quantum Free Electron Theory: 1. The energy of free electrons in a metal is quantized. 2. The electrons are distributed in a given energy level as per Pauli’s exclusion principle. (i.e., No two electrons will have the same set of four quantum numbers.) 3. Each Energy level can provide only two states namely; one with spin up and other with spin down. Hence, only two electrons can occupy in a given energy level. 4. The distribution of energy among the free electrons is according to Fermi-Dirac statistics. 5. The free electrons travel under a constant potential inside the metal and confined within material boundaries. 6. The attraction between the free electrons and lattice ions and the repulsion between valence electrons are ignored.  7. In this theory, though the energy levels of the electrons are discrete, the spacing between consecutive energy levels is very thin. Thus, the distribution of energy levels seems to be continuous. Blackbody Radiation Spectrum: Perfect Blackbody: An idealized physical body that absorbs all the incident electromagnetic radiation, regardless of frequency or angle of incidence, it will not reflect or transmit a radiation at low temperatures. The same black body emits different wavelengths of radiation at constant high temperatures. Blackbody Radiation: The heat radiation emitted from blackbody at constant high temperatures is known as blackbody radiation. The wavelength corresponding to maximum energy (or intensity) of emitted radiation depends only on the temperature of the blackbody and it does not depend on nature of the material emitting it. Construction of a Blackbody: In practice, a perfect black body is not available. The body showing close approximation to a perfect black body can be constructed. A hollow copper spherical shell is coated with lampblack on its inner surface. In this, a fine hole is made and a pointed projection is provided just in front of fine hole (Fig.1.1 (a)). The role of pointed projection is to prevent the reflected radiations to escape outside of the shell. When the heat radiations enter into this spherical shell through fine hole, the heat radiations suffer multiple reflections and they are getting completely absorbed. Now, this body acts as an absorber. When this body is placed at a constant high temperature, the heat radiations come out through fine hole (Fig. 1.1(b)). Now, this fine hole acts as a radiator not the walls of the body. (a) Absorber (b) Radiator Fig. 1.1 Blackbody Planck’s quantum theory of back body radiation: Max Planck introduced the revolutionary “Planck hypothesis” of black body radiation in the year 1900. This theory successfully explains the laws of black body radiation. Planck’s Theory (or) Planck’s radiation law: 1. A Black body is not only filled up with the radiations but also with a large number of tiny oscillators. They are of atomic dimensions. Hence, they are known as atomic oscillators or Planck’s oscillators. Each of these tiny oscillators vibrating with a characteristic frequency 2. The frequency of radiation emitted by oscillator is same as that of oscillator frequency. 3. The oscillator cannot absorb or emit energy in a continuous manner. It can absorb or emit energy in multiples of small units called quantum. This quantum of radiation is called photon. The energy of the photon (ε) is directly proportional to the frequency of radiation (ν), Where h known as Planck’s constant 4. The oscillator vibrating with frequency v can only emit energy in quantum of values hv. It indicates that the oscillators vibrating wit frequency v can only have discrete energy values En. En = n hv = n ε Where n is an integer (n = 1, 2, 3, 4…,), h is a Planck’s constant and is the frequency of the oscillating particle. According to Planck’s, the energy density emitted from a blackbody at a temperature T for all wavelength range λ and λ+dλ is given by, Limitation: According to this equation, the distribution of energy in blackbody radiation agrees well with the experimental observations in all wavelength range. Plank’s Radiation law: -----------------Eq. (1) i. For shorter wavelengths: Therefore, the value of is very large compared to ‘1’. Eq.(1) reduces to Assume C1=8πhc & C2=hc/K Therefore, C1 -----------------Eq. (2) Eq.(2) represents Wien’s radiation law. ii. For Longer wavelengths: Since, higher powers of smaller values can be neglected. -----------------Eq. (3) Eq.(3) represents Rayleigh-Jeans law. Photoelectric Effect: The emission of electrons from a metal plate when illuminated by light radiation of suitable wave length or frequency is called photoelectric effect. The emitted electrons are called photo electrons. This effect was discovered by Hertz, when ultraviolet light falls on zinc plate. This phenomenon was experimentally verified by the scientists, discovered that alkali metals like Li, Na, K etc. eject electrons when visible light falls on them. Millikan investigated this effect with a number of alkali metals over a wide range of light frequencies and was awarded Noble prize in 1923. The experimental arrangement to study the photoelectric effect is shown in figure. It consists of two photosensitive surfaces A and B enclosed in an evacuated quartz bulb. The plate A is connected to negative terminal of a potential device and plate B is connected to positive terminal through a galvanometer G or a micro ammeter. In the absence of light, there is no flow of current and hence there is no deflection in the galvanometer. When a monochromatic light is allowed to incident on plate A, current starts flowing in the circuit shown by galvanometer. The current is known as photo current. This shows that when light falls on the metal plate, electrons are ejected. The number of electrons emitted and their kinetic energy depends on 1) the potential difference between two electrodes i.e. between plate A and B 2) the intensity of incident radiation 3) the frequency of incident radiation 4) the photo metal used. Einstein’s Photo-electric equation: Following Planck’s idea that light consists of photons. Einstein proposed an explanation of photoelectric effect as early as 1905. According to Einstein’s explanation, in photoelectric effect one photon is completely absorbed by one electron, which thereby gains the quantum of energy and may be emitted from the metal.  The photons energy is used in the following two parts: (i) Apart of its energy is used to free the electron from the atom and away from the metal surface. This energy is known as photoelectric work function of the metal. This is denoted by W0. (ii) The other part is used in giving kinetic energy (1/ 2 𝑚𝑣 2) to the electron. Thus, hν = W0 + 1/ 2 𝑚𝑣 2 ---------- (1) Where ν is the velocity of emitted electron Eq. (1) is known as Einstein’s photoelectric equation. When the photon’s energy is of such a value that it can only liberate the electron from metal, then the kinetic energy of the electron will be zero. Eq. (1) now reduces to hν0 = W0 ----------- (2) Where ν0 is called the threshold frequency Threshold frequency is defined as the minimum frequency which can cause photoelectric emission. If the frequency of the photon is below threshold frequency no emission of electrons will take place. Corresponding to threshold frequency, we define long wavelength limit (λ0). It represents the upper limit of wavelength for photoelectric effect. Its physical significance is that radiations having wavelength longer than λ0 would not be able to eject electrons from a given metal surface whereas those having λ< λ0, will. Substituting the value of 𝑤0 = hν0 in equation (1), we have (3)  This is another form of Einstein’s Photoelectric equation. The Einstein’s photoelectric equation predicts all the experimental results. Thus, the increase in frequency ν of incident light causes increase in velocity of photoelectrons provided intensity of incident light is constant. An increase in the intensity of radiation is equivalent to an increase in the number of photons falling on the emitting surface. If the frequency of the incident radiation is above the threshold frequency ν > ν0, then the number of emitted electrons will increase. In this way the intensity of emitted electrons is directly proportional to the intensity of incident radiation. As h and e are constant 𝜗0 is also constant for a given photo cathode, eq. (7) shows that graph between stopping potential V0 and frequency ν would be straight line of slope h/e. Fundamental Laws of Photo-Electric Emission: 1. There is no time lag between incident radiation (photon) and ejected photoelectron. 2. The rate of photo-emission is directly proportional to intensity of incident radiation (light). 3. The velocity and hence the kinetic energy of photo-electrons is independent of intensity of incident light. 4. The velocity and hence the kinetic energy of photo-electrons is directly proportional to frequency of incident radiation. 5. The emission of electron take place above a certain frequency known as threshold frequency. This frequency is characteristic frequency of photo-metal used. Waves and Particles: De-Broglie suggested that the radiation has dual nature i.e., both particle as well as wave nature. The concept of particle is easy to grasp. It has mass, velocity, momentum, and energy. The concept of wave is a bit more difficult than that of a particle. A wave is spread out over a relatively large region of space, it cannot be said to be located just here and there, and it is hard to think of mass being associated with a wave. A wave is specified by its frequency, wavelength, phase, amplitude, intensity. Considering the above facts, it appears difficult to accept the conflicting ideas that radiation has dual nature. However, this acceptance is essential because radiation sometimes behaves as a wave and at other times as a particle. 1. Wave nature of radiation is observed in experiments based on interference, diffraction, polarization etc. 2. Particle nature of radiation is observed by blackbody radiation, Photoelectric effect, and Compton effect. However, radiation cannot exhibit both particle and wave nature simultaneously. De-Broglie’s Hypothesis: De-Broglie hypothesis states that like radiation, particles of matter also exhibit wave nature. The dual nature of matter was explained by combining Plank’s expression for the energy of a photon, E = h ν ……….Eqn.(1) with Einstein’s mass energy relation, E = m c2 ………. Eqn. (2) (Where c is velocity of light , h is Plank’s constant , m is mass of particle ) From Eqn.(1) & Eqn.(2) h ν = m c2………Eqn.(3) By introducing ν = c / λ, We get h c / λ = m c2 λ = h / mc λ = h / p ………Eqn.(4) Where p is momentum of a particle, c is the velocity of light in vacuum. λ is de-Broglie wavelength associated with a photon. De-Broglie proposed the concept of matter waves, according to which a material particle of mass ‘m’ moving with velocity ‘v’ should be associated with de-Broglie wavelength ‘λ’ given by λ = h / m v = h / p ………Eqn.(5) The above Eqn. represents de-Broglie wave equation. de-Broglie’s wavelength in terms of Energy: 1 mv 2 We know that the kinetic energy E= 2 Multiplying by m on both sides we get, 1 2 2 m v mE= 2 ………Eqn.(6) 2m E = m 2 v 2 (or) m 2 v 2 = 2m E Taking square root on both sides, mv = 2mE h We know that λ = 2mE ………Eqn.(7) By substituting mv value in Eqn.(7), we get h λ= 2mE This is de-Broglie wavelength in terms of Energy. de-Broglie’s wavelength associated with electrons: Let us consider the case of an electron of mass m and charge e, accelerated by a potential V volt from rest to velocity v. Then Energy gained by electron = eV 1 mv 2 Kinetic energy of electron = 2 1 mv 2 Therefore, eV = 2 2eV v= m By 12.26 0 A simplifying, we get λ = V. Therefore, when an electron accelerated through a potential difference of 100 V, the de-Broglie wavelength associated with it is 1.226 A°. The above equation can be applicable to all atomic particles like proton, neutron, and electron. Using this equation, we can calculate the wavelength associated with the material particles. Matter Waves: “The waves associated with the moving material particles are called matter waves”. Properties of Matter Waves: h We know that, de-Broglie wavelength, λ= mv 1. Lighter the particle, greater is the wavelength associated with it. 2. Smaller the velocity of the particle greater is the wavelength associated with it.  3. For v= o, λ= infinity, it means matter waves are associated only with the moving particles. 4. Whether the particle is charged or not, matter waves is associated with it. 5. It can be proved that matter waves travel faster that light. We know that E=h and E=mc2 mc 2  Therefore h = mc2 (or) h c2 Therefore phase velocity of matter wave is given by ω = λ = v As particle velocity ‘v’ cannot exceed the velocity of light, ω is greater than the velocity of light. i.e., matter waves can travel with a velocity greater than velocity of light. 6. The wave nature of matter introduces an uncertainty in locating the position and momentum of the particle. Since, a wave cannot said to be located at a particular point, which leads to the Heisenberg’s uncertainty principle. EXPERIMENTAL EVIDENCE FOR MATTER WAVES: 1. Davisson-Germer Experiment: We know that, waves exhibit diffraction phenomena. If the de Broglie hypothesis is valid, then the matter waves should exhibit diffraction effects. Diffraction is observed when the wavelength is comparable with the size of the object causing diffraction. Therefore, we anticipate the wave behaviour of an electron when it interacts with crystal. In 1927, Davisson and Germer observed the diffraction of an electron beam incident on a nickel crystal. The basic principle of their experiment is as the wavelength of an electron is of the order of X-rays, a beam of electrons must show diffraction effects from a crystal, like X-rays. The experiment provided a convincing proof of the wave nature of matter. Experimental arrangement:  Fig: 1.3 Schematic representation of Davisson-Germer Experiment The experimental setup of Davisson-Germer experiment is displayed in Fig. 1.3. It consists of electron gun, accelerating anode with pinholes, Nickel crystal, movable detector with sensitive galvanometer. The entire experimental setup was kept under vacuum. The role of electron gun is to produce a fine beam of electrons with required velocity. It consists of filament (F), low-tension battery (LTB), high-tension battery (HTB), and metal cylinder provided with pin holes. The heating of tungsten filament F by employing a low-tension battery produces large number of electrons. These electrons are accelerated to a required velocity by applying sufficient potential on metal cylinder (C) by employing a high-tension battery. The accelerated electrons are collimated into a fine beam of electrons as they pass through a system of pinholes provided in the cylinder C. The arrangement of target helps to get diffraction pattern. The target is a Nickel crystal. Beam of electrons with high velocity from the electron gun is made to incident on the nickel target. The incident electrons are reflected in all possible directions by the atomic planes of the crystal. An electron detector is fixed to a circular scale, which can collect the diffracted electrons from the crystal in different directions. The detector is connected to a sensitive galvanometer, which is useful to measure the intensity of electrons entering the detector. The intensity of the scattered electron beam was determined as a function of the scattering angle, ϕ.  Fig.1.4 Electric current Vs angle of scattering for various voltages A number of graphs were plotted between electric current and angle of scattering (angle between incident and diffracted electron beam) for electrons accelerated through various voltages (Fig. 1.4). It is found that for the accelerating voltage of 54 V, the electrons are scattered profoundly at an angle of 50° with the incident beam. Inter planar spacing of nickel crystal which is obtained from X-ray analysis, d= 0.91 A0. It is clear from the above figure that the glancing angle is. Applying Bragg equation, the ° wavelength of electron wave is computed as, 2d sin   n 2 × 0.91 × sinθ = nλ λ = 1.65 × 10-10 m (n =1, θ = 65o) λ = 1.65 Å The de Broglie wavelength associated with the electron, when a potential difference of 54 V is applied 12.27 12.27 Å Å λ= V = 54 = 1.667 Å  It is clear that the values obtained experimentally using Bragg’s equation and de Broglie equation are in good agreement. Hence, the wave nature of the particle is proved experimentally. There is a drawback in this experiment. We do not know exactly whether the observed diffraction pattern is due to electrons or due to electromagnetic radiation generated by fast moving electrons. Physical significance of (𝐰𝐚𝐯𝐞𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧): Born’s interpretation: The wave function 𝛙 is unable to give all possible information about the particle. 𝛙is a complex quantity and has no direct physical meaning. It is only a mathematical tool in order to represent the variable physical quantities in quantum mechanics. Born suggested that, the value of wave function associated with a moving particle at the position co-ordinates (x,y,z) in space, and at the time instant ‘t’ is related in finding the particle at certain location and certain period of time ‘t’. If 𝛙 represents the probability of finding the particle, then it can have two cases. Case 1: certainty of its Presence: +ve probability Case 2: certainty of its absence: - ve probability, but –ve probability is meaningless, Hence the wave function 𝛙 is complex number and is of the form a+ib Even though 𝛙 has no physical meaning, the square of its absolute magnitude |𝛙2 | gives a definite meaning and is obtained by multiplying the complex number with its complex conjugate then |𝛙2 |represents the probability density ‘p’ of locating the particle at a place at a given instant of time. And has real and positive solutions. 𝛙(𝐱, 𝐲, 𝐳,𝐭) = 𝐚 + 𝐢𝐛 𝛙∗ (𝐱, 𝐲, 𝐳,𝐭) = 𝐚 − 𝐢𝐛 𝐩 = 𝛙𝛙∗ = |𝛙2 | = 𝑎 2 + 𝑏 2 Where ‘P’ is called the probability density of the wave function If the particle is moving in a volume ‘V’, then the probability of finding the particle in a volume element dv, surrounding the point x, y, z and at instant ‘t’ is Pdv  ∫|𝛙2 |𝑑𝑣 = 1 𝑖𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑖𝑠 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 = 0 if particle does not exist This is called normalization condition. Heisenberg’s Uncertainty Principle: According to classical mechanics, a moving particle at any instant has a fixed position in space and a definite momentum, which can be determined simultaneously with any desired accuracy. The classical point of view represents an approximation, which is adequate for the objects of appreciable size, but not for the particles of atomic dimensions. Since a moving particle has to be regarded as a de-Broglie group, there is a limit to the accuracy with which we can measure the particle properties. The particle may be found anywhere within the wave group, moving with the group velocity. If the group is narrow, it is easy to locate its position but the uncertainty in calculating its velocity or momentum increases. If the group is wide, its momentum can be estimated satisfactorily, but the uncertainty in finding the location of the particle is great. According to Heisenberg’s uncertainty principle, “the simultaneous determination of exact position and momentum of a moving particle is impossible”. Qualitatively this principle states, “the order of magnitude of the product of the uncertainties in the measurement of position and momentum must be at least Planck’s constant h”. Where Δp is the uncertainty in determining the momentum and Δx is the uncertainty in determining the position of the particle, Similarly we have Where ΔE and Δt are the uncertainties in determining the energy. Schrodinger’s Time Independent Wave Equation: Schrodinger proposed a mathematical theory known as wave mechanics to describe the dual nature of matter. In 1926, Schrödinger derived a wave equation for moving particles by simply incorporating the de-Broglie wavelength expression into the classical wave equation. If a particle of mass ‘m’ moving with velocity ‘v’ is associated with a group of waves. Let ψ be the wave function of the particle. Also let us consider a simple form of progressing wave as represented by the following equation, Ψ = Ψ0 sin (ω t – k x) --------- (1) Where Ψ = Ψ (x, t) and Ψ0 is the amplitude Differentiating Ψ partially w.r.to x, ∂ Ψ/∂ x = Ψ0 cos (ω t – k x) (- k) = -k Ψ0 cos (ω t – k x) Once again differentiate w.r.to x ∂2ψ / ∂x2 = (- k) Ψ0 (- sin (ω t – k x)) (- k) = - k2 Ψ0 sin (ω t – k x) ∂2 ψ / ∂x2 = -k2 ψ (From Eqn.1) ∂2 ψ / ∂ x2 + k2 ψ = 0 ----------- (2) ∂2 ψ / ∂ x2 + (4 Π2 / λ2) ψ = 0 --------- (3) (Since k = 2 Π / λ) Eqn. (2) or Eqn. (3) represents the differential form of the classical wave eqn. Now we incorporate de-Broglie wavelength expression λ = h / m v. Thus, we obtain ∂2 ψ / ∂ x2 + (4 Π2 / (h / m v)2) ψ = 0 ∂2 ψ / ∂ x2 + (4 Π2m2 v2 / h2) ψ = 0 -------------- (4) The total energy E of the particle is the sum of its kinetic energy K and potential energy V i.e., E = K + V -------------- (5) 1 m & K = 2 v2 ---------- (6) Therefore m2 v2 = 2 m (E – V) ------------ (7) From (4) and (7), ∂2 ψ / ∂ x2 + [8 Π2 m (E-V) / h2] ψ = 0 ------------ (8) In quantum mechanics, the value h / 2 Π occurs more frequently. Hence, we denote ђ=h/2Π Using this notation, we have ∂2 ψ / ∂ x2 + [2 m (E – V) / ђ2] ψ = 0 ------------ (9) For simplicity, we considered only one – dimensional wave. Extending Eqn. (9) for a three – Dimensional wave, we have ∂2 ψ / ∂ x2 + ∂2 ψ / ∂ y2 + ∂2 ψ / ∂ z2 + [2 m (E – V) / ђ2] ψ = 0 ------------ (10) Where Ψ = Ψ (x, y, z). Here, we have considered only stationary states of ψ after separating the time dependence of Ψ. Using the Laplacian operator, ▼2= ∂2 / ∂ x2 + ∂2 / ∂ y2 + ∂2 / ∂ z2 ------------- (11) Eqn. (10) can be written as ▼2 Ψ + [2 m (E – V) / ђ2] ψ = 0 --------------- (12) This is the Schrödinger Time Independent Wave Equation. Why electrons cannot exist inside the nucleus (application of uncertainty principle): The radiation emitted by radioactive nucleus consists of α, β, γ out of which β-rays are identified to be electrons. We apply uncertainty principle whether electrons are coming out of the nucleus. The radius of the nucleus is of the order of 10-14m. Therefore, if electrons were to be in the nucleus, the maximum uncertainty Δx in the position of the electron is equal to the diameter of the nucleus. Thus, Δx= 2*10-14 m The minimum uncertainty in its momentum is then given by, h 6.62 *10 34 Js p    5.2 *10 21 kg  m / s 2 * x 2 * 3.14 * 2 *10 m 34 The minimum uncertainty in momentum can be taken as the momentum of the electron. Thus, p = 5.2 x 10-21 kg-m/s. The minimum energy of the electron in the nucleus is given by, Emin = pmin * c= (5.2 x 10-21 kg-m/s)(3 x 108 m/s)= 1.56*10-12J=9.7 MeV. It implies that if an electron exists within the nucleus, it must have a minimum energy of about 10 MeV. However, the experimental measurements showed that the maximum kinetic energies of β-particles were of the order of 4 MeV only. That means no electron or particle in the atom possess energy greater than 4MeV. From this, it can be concluded that electrons are not present in the nucleus. Particle in One Dimensional Box (Infinite Square Well): Consider the case of a particle of mass ‘m’, which is bound to move in a one-dimensional potential well of infinite height and width a. Fig. 1.6 Particle in a potential well of infinite height The particle is free to move inside the well and hence the potential energy of the particle is assumed zero. At the same, the potential energy of the particle outside the walls is infinite due to the infinite potential energy outside the potential well. Thus, we have V(x) = 0 for 0 < x < a V(x) =  for x ≤ and x ≥ a The Schrodinger time independent wave equation for the particle is, d 2 2m  2 ( E  V )  0 dx 2 Since for a free particle V=0, d 2 2mE  2  0 dx 2  d 2 2mE  K 2   0 2 dx 2 ---------- (1), Where K2 = The solution of the above equation can be written as,  ( x)  A sin Kx  B cos Kx ---------- (2) Where A and B are constants. The values of these constants can be obtained by applying boundary conditions given by At x=0;  0 At x=a;   0 Applying first boundary condition for Eqn. (2), we have 0=Asin0+Bcos0 Implying, B=0 (Since Cos90°=1)  ( x)  A sin Kx ---------- (3) Applying second boundary condition for (3) we have 0 = A sinKa  SinKa  0 (or) Ka= n π, Where n = 0, 1, 2,…… n K a Now Eqn. (3) becomes nx   x   A sin a ----------- (4) n  2 n 2 2 K K  2 Since a a and we have 2mE n 2 2  2  a2 (Using En for E, for different n values we will get different E values)  n 2 2 2 n 2 h 2 En   2ma 2 8ma 2  En  n 2 This shows that energy of particle is quantized. The discrete energy values are given by 2 2 E1  2ma 2 , for n=1 4 2 2 E2   4 E1 2ma 2 , for n=2 9 2 2 E3   9 E1 2ma 2 , for n=3 16 2 2 E4   16 E1 2ma 2 , for n= 4 and so on. The constant A can be obtained by applying normalization condition, i.e., a   ( x) dx  1 2 0 a nx A dx  1 2 sin 2 0 a a nx A 2  sin 2 dx  1 0 a a 1 2nx  A 2  1  cos dx  1 0 2 a  a A2  a 2nx   x  sin 1 2 2n a  0 A2 a 1 2 2 2 A2  A a or a  2  n   n ( x)  sin  x a  a  The wave function ψ1 has two nodes at x = 0 & x = a The wave function ψ2 has three nodes at x = 0, x = a/2 & x = a The wave function ψ3 has three nodes at x = 0, x = a/3, x = 2a/3 & x = a The wave function ψn has (n+1) nodes 0 a 0 a x x Fig. 1.7 Representation of Wave function and Probability densities of a freely moving particle in 1-D box at different energy levels