OCR Part 2A PDF

Part 2A 1 Outline 2.1 The behavior of wave at the boundary 2.2 Confining a particle 2.3 Schrodinger Equation : Free particle (1D) 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization (1D)...

Part 2A 1 Outline 2.1 The behavior of wave at the boundary 2.2 Confining a particle 2.3 Schrodinger Equation : Free particle (1D) 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization (1D) 2 2.1 Behavior of wave at the boundary In classical physics (nonrelativistic, nonquantum): In nonrelativistic quantum mechanics: The behavior of a particle can be explained according The basic equation to be solved is a second-order differential to Newton’s laws. equation known as the Schrodinger equation. 1) Particle interaction: describe in terms of force. 1) Particle interaction: describe in terms of potential energy (rather than the force). 2) According to Newton’s law, some relationships of 2) Schrodinger Eq.’s solution gives the wave function of the particle, e.g. location (x) and velocity (v), can be particle, which carries information about the behaviors of wave obtained. particle. 3) F = ma, predicts how an object’s behavior changes Example of information: - The probability density of finding the particle due to applied forces. - The solutions of the time-independent Schrödinger Eq. for electrons in an atom correspond to quantized energy levels. - Quantum tunneling of a particle that tunnels through an energy barrier that would be classically impossible. 3 Note 1: Wave function of the particle carries information about the behaviors of wave particle. For example: Free Electrons: Not bound to any atom and is not under the influence of external forces (fields). The wavefunction of a free e- in 1D can be described by solution of the time-independent Schrödinger equation for a free particle. (no potential acting on the electron) (angular wavenumber) -> 2π comes from expressing the wave's oscillation in terms of right-moving left-moving radians. -> Unit, e.g., radians per meter. -> wavenumber without the angular (radian) factor. -> Unit - e.g., cycles per meter. 4 2.1 Behavior of wave at the boundary 𝜆" = 𝜆# 𝐴" < 𝐴# de Broglie waves of electrons Light wave in air Surface wave a) The wavelength (𝜆) in air in a) Electrons moving from ‘region 1’ of a) In shallower depth, its region 3 is the same as the original constant zero potential to ‘region 2’ of wavelength is smaller, but its wavelength of the incident wave in constant negative potential 𝑽𝟎 also have amplitude is larger. region 1. transmitted and reflected components. b) At region 3 (the same depth as in b) The amplitude (A) in region 3 is 1), the λ returns to its original value, b) At ‘region 3’, =? less than the amplitude in region 1, but the amplitude of the wave in 3 is because some of the intensity is smaller (some intensity was reflected). ‘ we focus on here.’ reflected at A and at B. (Note: Amplitude of Tsunami Propagation) 5 2.1 Behavior of wave at the boundary 1) In region 1, electrons move inside a narrow metal rod that is at ground potential (V = 0). (Assume one electron first) 2) Region 2 is connected to the negative terminal of a battery, keeping it at uniform potential of −𝑉$. 3) Region 3 is connected to region 1 at ground potential. 4) The gaps between the tubes be made so small that we can regard the changes in potential at A and B as occurring suddenly. 𝒉 In region 1: Electron has ’kinetic energy (𝑲)’, ‘momentum (𝒑 = 𝟐𝒎𝑲)′ and ‘de Broglie wavelength (𝝀 = 𝒑)’. In region 2: Potential energy for the electron is 𝑼 = 𝒒𝑽. (Assume that K in region 1 is higher than U) Electrons move into In region 2 with a smaller kinetic energy, smaller momentum, and thus greater wavelength. (𝑝 ∝ 𝐾) In region 3: Electron gains back the lost kinetic energy and moves with its original kinetic energy K and thus with their original wavelength. Similar to the light wave or the water wave, the amplitude of the de Broglie wave in 𝜆" = 𝜆# region 3 is smaller than in region 1 because some of the electrons are reflected. 𝐴" < 𝐴# 6 2.1 Behavior of wave at the boundary In region 2: Potential energy for the electron is 𝑼 = 𝒒𝑽 𝑒' Like light waves, at suitable (𝑲 > 𝑼), electron can go through the forbidden region with decreasing amplitudes. 𝜆" = 𝜆# 𝐴" < 𝐴# Suppose we increase the battery voltage so that the potential In some case, the penetration of the electrons (wave- energy in region 2 is greater than the initial kinetic energy like behavior) into the forbidden region is also possible in region 1 𝑲 < 𝑼. and is related to the uncertainty principle, i.e., the probability of tunneling events occurring What would happen ?. 7 2.1 Behavior of wave at the boundary Continuity at the Boundaries When a wave, e.g. light, water surface wave, matter wave, crosses a boundary, The mathematical function that describes the wave must have two properties at each boundary: 1. The wave function must be continuous. 2. The slope of the wave function must be continuous, except when the boundary height is infinite. A discontinuous wave function (not allowed) Sine curve (c) and exponential curve (d): Continuous wave function with a both wave function and discontinuous slope. slope are continuous. (not allowed, except the boundary is of infinite height. ) 8  1) Wave function is continuous at boundary A. 2.1 Behavior of wave at the boundary Thus, wave fnc. In region 2 should be From the Fig., the wave particle in ‘region 1’ is 𝑦( 𝑥 = 𝐶( sin ()* − 𝜙( (1) +! 𝑦# = 𝐶# sin(2𝜋𝑥/𝜆# − 𝜙# ) At the boundary A (x = 0) 𝐶# = 11.5, 𝜆# = 4.97 cm, 𝜙# = −65.3° 𝑦# 𝑥 = 𝑦( 𝑥 In ‘region 2’, 𝜆( is 10.5 cm. the boundary ‘A’ is at x = 0 cm At the boundary ‘B’ is at x = L = 20 cm 𝐶# sin(−𝜙# ) = 𝐶( sin(−𝜙( ) Find 𝜙( and 𝐶( =? −𝐶# sin(𝜙# ) = −𝐶( sin(𝜙( ). (2) 2) The slope in region 1 match the slope in region 2 at boundary A. 𝑖𝑓 𝑦 = Csin(2𝜋𝑥/𝜆 − 𝜙) The slopes à dy/dx = (2π/λ)C cos(2πx/λ − φ) At the boundary A (x = 0) X =0 X =L !" !" Sol 𝐶 cos(𝜙$ ) = 𝐶 cos 𝜙!. (3) #! $ #" ! Use two boundary conditions. 9 2.1 Behavior of wave at the boundary −𝐶# sin(𝜙# ) = −𝐶( sin(𝜙( ). (2) To find 𝑪𝟐 () () From Eq.(2) 𝐶 cos(𝜙# ) +" # = 𝐶 cos +! ( 𝜙(. (3) sin(𝜙# ) 𝐶( = 𝐶# sin(𝜙( ) Eq.(2)/Eq.(3) sin(−65.3) #! #" 𝐶( = (11.5) 𝑡𝑎𝑛 𝜙$ = 𝑡𝑎𝑛 𝜙! sin(−45.8) !" !" 𝜆$ 𝑡𝑎𝑛 𝜙$ = 𝜆! 𝑡𝑎𝑛 𝜙! 𝑪𝟐 = 𝟏𝟒. 𝟕 𝜆$ From Eq.(1) 𝑡𝑎𝑛 𝜙! = 𝑡𝑎𝑛 𝜙$ 𝜆! %$ 𝜆$ Thus, 𝑦! 𝑥 = 𝐶! sin( !"( − 𝜙! ) 𝜙! = 𝑡𝑎𝑛 ( 𝑡𝑎𝑛 𝜙$ ) #" 𝜆! 4.97 𝜙! = −45.8) 𝜙! = 𝑡𝑎𝑛%$ ( 𝑡𝑎𝑛 −65.3 ) 10.5 𝐶( = 14.7 10 𝝓𝟐 = −𝟒𝟓. 𝟖𝒐 2.2 Confining a particle The elements of an idealized “trap” designed Free particle => a particle with no external forces (field) acting on to confine an electron to the central cylinder.. => not confined and can be located anywhere. (Negative) (Negative) (if moving, it is under its own momentum and energy as originated from the initial condition) Note: a free particle is a useful idealization that simplifies theoretical analyses and helps in understanding the fundamental properties of particles. − 𝑉$ − 𝑉$ Confined particle => In QM, a particle is constrained within a specific region of space due to the presence of a potential energy barrier or well. This confinement leads to quantized energy levels. Example: Electron is bound to the nucleus. The electron's energy levels are quantized, meaning they can only take on certain discrete values. = (−𝑒)(−𝑉! ) L a potential energy well 11 2.2 Confining a particle To confine the electron, electron’s kinetic energy 𝑲 < 𝑼𝟎 in the center section. For example, The side sections have U = 𝑈$ = 10 eV. If electron has ‘K’ of less than 10 eV, it does not have enough energy to ‘escape’ the walls. In classical physics, electron behaves like a particle. Thus, the probability to find the electron out of the potential well is zero. L How about in the view of QM ? 1. In the center, the potential energy (U) = 0. 2. In the sides, 𝑈 = 𝑈* = 𝑞𝑉 = −𝑒 −𝑉* 12 2.2 Confining a particle Based on standing wave concept In QM, an infinite potential well is a theoretical example where a particle is The standing wave concept can be generalized confined within a region of space by infinitely to various scenarios where particles are high potential energy barriers at its constrained in their motion by external boundaries. factors, leading to discrete energy states. Here, the Infinite potential well is one of these examples. The potential energy (U) inside the well is zero, the total energy is equal to the kinetic Thus, are the energy levels of the wave-like particle energy. quantized? 𝐸HIHJK = 𝐾 + 𝑈 Infinite potential well 13 Note 2: (PHY 101/PHY191) 2𝐿 𝑣 Standing waves 𝜆= 𝑓 =𝑛 𝑛 2𝐿 𝑣 𝜆! = 2𝐿 , 𝑛 = 1 𝑓! = ,𝑛 = 1 2𝐿 2𝐿 𝜆! 𝑣 𝜆" = = 𝑓" = 𝑛 = 2𝑓! 2 2 2𝐿 2𝐿 𝜆! 𝑣 𝜆# = = 𝑓# = 𝑛 = 3𝑓! 3 3 2𝐿 2𝐿 𝜆! 𝑣 𝜆$ = = 𝑓$ = 𝑛 = 4𝑓! 4 4 2𝐿 Note: n is the number of antinode. it may imply the number of loop. 2.2 Confining a particle Based on standing wave concept Free particle à Wavelength could ∞ ∞ have any value (𝜆 = ℎ/𝑝). Confined particle à certain values of 𝑈=0 the wavelength are allowed. Here, L energy is ‘not’ continuous. It is discretes, called ‘Quantization of energy’. 𝜆! = 2𝐿 The allowed wavelengths in the potential well are 2L, L, 2L/3,..., where L is the length of the center 𝜆! = 2𝐿/2 section. 𝜆! = 2𝐿/3 n = quantum number (Eq. 1) 𝜆! = 2𝐿/4 15 2.2 Confining a particle Based on standing wave concept Assume that the energy of electron in the center is only kinetic energy. (U= 0) From 1 1 𝑚 𝐾= 𝑚𝑣 " = 𝑚𝑣 " ( ) 2 2 𝑚 and i (kl)( = j k i m( K= jk then 2𝐿 ℎ ! " = then T (S ) 𝑛 𝑝h 𝐾S = "# U V #! 𝐸! = 𝑛" $%&! (Eq. 2) These are the allowed or quantized (Eq. 3) values of the energy of the16 electron (infinite potential well from standing wave concept ) at L=100 pm. 2.3 Schrodinger Equation Schrodinger equation : Differential equation : Its solution gives us the wave function of a particle. Dimension of moving wave-like particle (Cartesian, polar and spherical coordinates) 1 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 1𝐷 , 𝑒. 𝑔. 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑚𝑜𝑣𝑖𝑛𝑔 𝑎𝑙𝑜𝑛𝑔 𝑥 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 −→ the wave function is ψ(x). 2 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 2𝐷 , 𝑒. 𝑔. 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑚𝑜𝑣𝑖𝑛𝑔 𝑎𝑙𝑜𝑛𝑔 𝑥, 𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠 −→ the wave function is ψ(x, y). 3 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 3𝐷 , 𝑒. 𝑔. 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑚𝑜𝑣𝑖𝑛𝑔 𝑎𝑙𝑜𝑛𝑔 𝑥, 𝑦, 𝑧 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠 −→ the wave function is ψ(x, y, z). Schrodinger equation : 1) 𝑇𝑖𝑚𝑒 − 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝐸𝑞. −→ the wave functions are ψ(x); ψ(x, y);ψ(x, y, z). 2) 𝑇𝑖𝑚𝑒 − 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝐸𝑞. −→ the wave functions are ψ(x, t); ψ(x, y, t);ψ(x, y, z, t). 17 2.3 Schrodinger Equation : Free particle Schrodinger equation : Differential equation : Its solution gives us the wave function of a particle. Time-independent Schrodinger equation for one-dimensional motion. 𝑑j𝜓 𝑥 j𝜓 𝑥 , 2𝑚(𝐸 − 𝑈) = −𝑘 𝑘= 𝑑𝑥 j ℏj or ℏj 𝑑 j 𝜓 𝑥 𝑘 = wavenumber − j + 𝑈 𝑥 𝜓 𝑥 = 𝐸𝜓 𝑥 2𝑚 𝑑𝑥 𝐸 = 𝑇𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑈 = 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 Its solution gives the shape of the wave for time independent Eq. 𝑚 = 𝑚𝑎𝑠𝑠 18 Note : Wave function of the particle carries information about the behaviors of wave particle. For example: Free Electrons: Not bound to any atom and is not under the influence of external forces (fields). The wavefunction of a free e- in 1D can be described by solution of the time-independent Schrödinger equation for a free particle. (no potential acting on Hamiltonian operator the electron) Potential energy operator right-moving left-moving kinetic energy operator 19 2.3 Schrodinger Equation : Free particle Schrodinger equation for Free particle z z z p= = () = 𝑘 = ℏ𝑘 From Change the ‘k’ term to be ‘energy’ term { (*) j| p = ℏ𝑘 𝑑j𝜓 𝑥 f" ℏ" h " = −𝑘 j𝜓 𝑥 K= = 𝑑𝑥 j UV UV What do you see inside ‘𝒑 = ℏ𝒌’ ? 2𝑚(𝐸 − 𝑈) 𝑘= = wave-particle duality, which is ℏ! UVi UVj the concept of modern physics. U Note: 𝑘 = ℏ" = ℏ" 𝑘 = wavenumber What do you see inside ‘𝒑 = 𝒎𝒗’ ? 𝑲 = 𝒌𝒊𝒏𝒆𝒕𝒊𝒄 𝒆𝒏𝒆𝒓𝒈𝒚 = particle behavior, which is the concept of classical physics. For a free particle, U = 0 so E = K 𝑑j𝜓 𝑥 2𝑚𝐾 (𝐸 = 𝐾 + 𝑈) j =− j 𝜓 𝑥 𝑑𝑥 ℏ Note: A free particle is not subject to any external forces or potentials, so there is For a free particle no potential energy associated with its motion. 20 2.3 Schrodinger Equation : Free particle Schrodinger equation for Free particle Schrodinger equation (differential form): Time independent and 1D movement. 𝑘 = wavenumber 𝑑"𝜓 𝑥 2𝑚(𝐸 − 𝑈) " = −𝑘 " 𝜓 𝑥 , 𝑘= 𝑑𝑥 ℏ" 𝑬 = 𝑻𝒐𝒕𝒂𝒍 𝒆𝒏𝒆𝒓𝒈𝒚 𝑼 = 𝑷𝒐𝒕𝒆𝒏𝒕𝒊𝒂𝒍 𝒆𝒏𝒆𝒓𝒈𝒚 𝑑j𝜓 𝑥 2𝑚𝐾 j =− j 𝜓 𝑥 𝒎 = 𝒎𝒂𝒔𝒔 𝑑𝑥 ℏ For free particle, the solution can be specified by the wave function ψ(x). The possible solutions are below (ขึน# กับเงื+อนไข, condition): 𝝍 𝒙 = 𝑨 𝒆𝒊𝒌𝒙 𝝍 𝒙 = 𝑨 𝒆𝒊𝒌𝒙 + 𝑩𝒆(𝒊𝒌𝒙 21 2.3 Schrodinger Equation: Free particle Schrodinger equation for Free particle 𝑇𝑜𝑡𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝐹𝑟𝑒𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒(𝐸) = ? Method: 2 Method: 1 1 " 1 " 𝑚 𝐸 = 𝐾 + 0 = 𝑚𝑣 = 𝑚𝑣 ( ) 2 2 𝑚 From Time-independent Schrodinger Eq., i (kl)( 𝑑"𝜓 𝑥 2𝑚 = j k = − " (𝐸 − 𝑈)𝜓 𝑥 ………………..(1) 𝑑𝑥 " ℏ 1 𝑝" ℏ" 𝑘 " = = 𝑑"𝜓 𝑥 "𝜓 𝑥 , 2𝑚(𝐸 − 𝑈) 2𝑚 2𝑚 = −𝑘 𝑘= ……..(2) 𝑑𝑥 " ℏ" The energy of the free particle is 2𝑚 𝑘j = j 𝐸 ,𝑈 = 0 ℏj 𝑘 j ℏ E= Ans. 2𝑚 ℏj j Here, for free particle, the energy is not quantized. 𝐸= 𝑘 (different from confined particle) 22 2𝑚 Ans. 2.3 Schrodinger Equation Schrodinger equation for Free particle: Time dependent wave function Note: For traveling wave, the mathematical function that describes a 1D direction must involve both The time dependence is given by x and t. This wave is represented by the 1. 𝜔𝑡 (sine- cosine form) function (x, t): 2. 𝑒 ()*+ (complex exponen@al form) with 𝝎 = 𝑬/ℏ. (Quantum system) 𝑬 Prove that 𝝎 = 𝒊𝒔 𝒄𝒐𝒓𝒓𝒆𝒄𝒕𝒆𝒅. ℏ Example, 𝐸 = ℎ𝑓 (general for E of photons and electromagne[c waves) 𝜓 𝑥, 𝑡 = 𝜓(𝑥) G 𝑒 ‚ƒ„… 𝝎 𝐸 = ℎ( ) , 𝜔 = 2𝜋𝑓 𝜓 𝑥 = 𝑒 ƒ†‡ 𝟐𝝅 ℎ 𝐸 = ( )𝝎 𝟐𝝅 𝐸 = ℏ𝝎 (general for E of quatum particle in the quantum system) 23 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Where is the particle in the space, e.g. bound electron ? = We need to know ‘probability density’ first. Probability density ! Note: in this class, we focus on 1D + time independent. 𝜓(𝑥) 𝜓(𝑥, 𝑡) ! Example of complex system 1. Real and non-negative quantity. 2.The total value over all positions must equal 1, (Probability of detection). 3. Higher |ψ(x)|2 value corresponds to higher probability density. 24 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Probability Density (Time independent ) Example: Assume that it is a free particle 𝜓 𝑥 = A𝑒 )(/0) => probability density (per unit length) 𝜓(𝑥) " = 𝜓∗ „ 𝜓 => Squared absolute amplitude 𝜓(𝑥) ! gives the probability for finding the particle at a small region of space (x). = A𝑒 ()(/0) „ A𝑒 )(/0) 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦: 𝜌(𝑥) = 𝜓(𝑥) ! = 𝐴" (𝑒 ()(/0) „ 𝑒 )(/0 ) ! ∗ = 𝐴" 𝜓(𝑥) =𝜓 𝜓 𝜓*(x) is the complex conjugate of the wave function. Here, the plot of probability density of ‘a free particle’. ! 25 𝜓(𝑥) is constant. 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Probability Density (Time dependent ) ! 𝜓(𝑥, 𝑡) = 𝜓∗𝜓 𝜓*(x,t) is the complex conjugate of the wave function. Example: Assume that it is a free particle 𝜓 𝑥, 𝑡 = A𝑒 )(/0(*+) 𝜓(𝑥) " = 𝜓∗ „ 𝜓 = A𝑒 ()(/0(*+) „ A𝑒 )(/0(*+) = 𝐴" (𝑒 ()(/0(*+) „ 𝑒 )(/0(*+ ) = 𝐴" 26 Note : Probability density (𝜌(𝑥)) : finding a particle in a Probability of detection 𝑷 𝒙𝟏, 𝒙𝟐 particular region of space. For free particle, -> Prob. of detecting the particle within a specific interval x1 and x2 -> calculated by integrating the probability density -> Probability density associated with finding over that interval. the particle at position x. -> It has units of probability per unit length (e.g., m−1) for 1D. à a dimensionless quantity, ranging from 0 to 1. 27 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Probability of detection 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 = 𝜓(𝑥) ! 1. Probability of detecting a particle in the width ′𝑑𝑥′ or in between x and x+ dx 2. Probability 𝒐𝒇 detecting a particle in between 𝑥! 𝑎𝑛𝑑 𝑥" is. ‡( 𝑷 𝒙𝟏 , 𝒙𝟐 = J 𝜓(𝑥) j𝑑𝑥 ‡+ 𝑥! 𝑥" 28 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Wavefunction for a Free Particle (x direction): Probability of Detection Probability of Detecting the Particle in a Specific Interval. Suppose we want to find the probability of detecting the particle in the interval [x1,x2]=[0,1] m Probability Density supported This indicates that the particle has an equal probability of being found anywhere along the x-axis. 29 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Particle in an Infinite Potential Well (see Slide 15) Length of the well, L = 1 m First energy level (n=1) wavefunction Probability of Detection Calculation Q: Probability of detecting the particle in the interval [0.25,0.5]. Probability Density Calculation 𝑃 0.25, 0.5 30 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Probability Densities Probability of Detection in the interval [0.25,0.5] The probability density gives a measure of at a specific point, while the probability of detection provides measurement over an interval, which is always between 0 and 1. 31 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Normalization Normalization => the total probability of detection = 1 Normalization equation J 𝜓(𝑥) j𝑑𝑥 = 1 – J 𝜓(𝑥) j𝑑𝑥 = 1 Note: Normalization ensures that the total ”“ probability is equal to 1 (finding a J 𝜓(𝑥) j𝑑𝑥 = 1 particle at any position sum up to ‚“ 100%). Confirming that the particle must exist somewhere in space. 32 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Consider a one-dimensional infinite potential well (particle in a box) with width L. The wave function of the From, particle in the nth energy state is : Q: Verify whether this wave function is normalized. To verify the normalization of the wave function 𝜓 𝑥 is normalized, meaning the total probability of finding 33 the particle somewhere in the interval from 0 to L is 100%. 2.4 Schrodinger Equation: Probability Density, Probability of detection and Normalization Wave function for a particle in a one-dimensional potential well : where A is the normalization constant (amplitude) and α > 0 is a constant that defines the exponential decay. Q: A = ? 34