Explain ---```plain $-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V\psi = i\hbar\frac{\partial\psi}{\partial t}$ (in one dimension) $-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = i\hbar\frac{\partial\psi}{\partial t}$ (in three dimension) 2) Time independent equation. $-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V\psi = E\psi$ (in one dimension) $-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi$ (in three dimension) Where $\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} = \nabla^2$ represents differential operator (Laplace Operator) Q. Show that wave function for a particle confined in an infinite one dimensional potential well of length “L” is given by$\psi_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi}{L}x)$. Hence, discuss the energy levels and their discreteness. OR Q. Show that wave function for a particle confined to move in infinite one dimensional potential well of length (L) is given by $E_n = \frac{n^2h^2}{8mL^2}$ where symbols have their usual meanings. Is the electron trapped in a potential well, allowed to take zero energy ? Why ? Q. Show that the energy of an electron confined in one dimensional potential well of length (L) and infinite depth is quantized? Is the electron trapped in a potential well, allowed to take zero energy? Why ? or Q. A free particle of mss “m” is kept in a rectangular box of length “L”. Considering one dimensional motion, obtain an expression of discrete energy of particle. Show that energy of particle are quantized. or Q. A particle confined in a one dimensional potential well of infinite depth. Use Schrödinger's wave equation to obtain the energy states of a particle inside the well. Or Q. Write down Schrodinger’s time-independent wave equation for matter waves. Hence obtain an expression for the energy of a particle in one dimensional potential well of infinite height, Infinite potential well OR Particle in a potential well OR Particle in a Rigid box A potential well is a potential energy function V(x) that has a minimum energy. If a particle is left in ``` **Textual Information:** the well and the total energy of the particle is less than the height of the potential well, then the particle is said to be **trapped** in the well. In classical mechanics a particle trapped in a potential well can vibrate back and forth with periodic motion but cannot leave the well. In quantum mechanics, such trapped state is called a **bound state**. Let us consider a Particle of mass 'm' moving with velocity v along positive x-direction. Let its motion be restricted between x = 0 and x=L inside the box bounded by rigid walls. The Particle bounces back and forth on the wall of rigid box. When it collides with the walls it does not lose any energy as the collision being elastic collision. And hence total energy E remains constant. Now the P.E of particle outside the box is infinite and inside the box it is constant but for our convenience inside the box constant P.E.is zero. Thus, v_x = ∞, for 0 > x > L v_x = 0, for 0 ≤ x ≤ L The variation of potential with x is shown in fig. The particle is inside in deep potential well. **Schrödinger's time independent wave equation is** Let ψ be the wave function associated with it. The wave function ψ = 0 outside the box because the particle exist inside the box. To find the energy and wave function inside the box we apply Schrödinger's time-independent equation 1- Dimensional h^2 d^2ψ / - 2m dx^2 + Vψ = Eψ The particle is moving along x-direction. - h^2 d^2ψ / 2m dx^2 + (E-V)ψ = 0 The potential energy v = 0 inside the box. ∴ d^2ψ / dx^2 + 2mEψ / h^2 = 0 Keep 2mE / h^2 = k^2 ------------ (1) d^2ψ / dx^2 + k^2ψ = 0 **Chart/Diagram Description:** * **Type:** Potential well diagram / 1-Dimensional Box Potential. * **Main Elements:** * **Coordinate Axes:** Vertical axis labeled 'V' (Potential), horizontal axis labeled 'x' (Position). An upward arrow on the V axis indicates the direction of increasing potential. * **Lines:** Two vertical black lines representing potential walls at x = 0 and x = L. The lines extend infinitely upwards, indicating infinite potential outside the box. A horizontal line segment along the x-axis between x=0 and x=L indicates zero potential inside the box. * **Points:** A red circle is located on the horizontal line segment (V=0) somewhere between x=0 and x=L, representing the particle inside the potential well. * **Labels:** * 'V' label with an arrow indicating the potential axis. * 'x=0' label below the left wall. * 'x=L' label below the right wall. * 'Ψ=0, V≠0' labeled to the left of the left wall, likely indicating the wave function is zero and potential is non-zero (infinite) outside. * 'V=0, Ψ≠0' labeled between the walls, likely indicating the potential is zero and the wave function is non-zero inside. * 'Ψ=0, V≠0' labeled to the right of the right wall, likely indicating the wave function is zero and potential is non-zero (infinite) outside. * **Relative Position and Direction:** The two vertical walls are parallel and perpendicular to the x-axis. The particle is depicted as being contained within the region between the walls at x=0 and x=L where the potential is zero. The general solution of this equation is ------------------------------------------------- (2) $\psi = A \sin kx + B \cos kx$ ------------------------------------------------- (3) Where A & B are constant and its value can be obtained by applying boundary conditions i.e. at x = 0 => $\psi = 0$ at x = L => $\psi = 0$ Now applying first condition at x = 0 => $\psi = 0$ eq$^{n}$.(3) => $0 = A \sin 0 + B \cos 0$ B = 0 $\because \cos 0 = 1$ Put in eq$^{n}$.(3), we get $\psi = A \sin kx$ ------------------------------------------------- (4) at x = L => $\psi = 0$ eq$^{n}$.(3) => $0 = A \sin kL + 0$ $\therefore A \neq 0 \Rightarrow \sin kL = 0 \Rightarrow kL = n\pi \Rightarrow k =$ $k = \frac{n\pi}{L}$ ------------------------------------------------- (5) $K = \frac{2\pi}{\lambda} \Rightarrow \frac{2\pi}{\lambda} = \frac{n\pi}{L} \Rightarrow \lambda = \frac{2L}{n}$ This implies that the wave equation has solution only when the electron wavelength is restricted to discrete values such that only a whole number of half wavelengths $(\lambda/2)$ is formed over the length L. It means that electron waves form standing wave pattern within the potential well. From eq$^{n}$. (1) & (5) We get, $\frac{2mE}{\hbar^2} = k^2$ $\frac{2mE}{\hbar^2} = \frac{n^2\pi^2}{L^2}$ $E_n = \frac{\hbar^2 n^2\pi^2}{2mL^2}$ $= \frac{2mL^2}{n^2\pi^2 h^2}$ (This line appears to be an incorrect algebraic step in the source image) $= \frac{n^2 h^2}{8mL^2}$ $E_n = \frac{n^2 h^2}{8mL^2}$ ------------------------------------------------- (6) Where n = 1,2,3---------- For n = 1 E1 = h^2 / (8mL^2) For n = 2 E2 = 4h^2 / (8mL^2) For n = 3 E3 = 9h^2 / (8mL^2) For n = 4 E4 = 16h^2 / (8mL^2) ... are allowed energy states or energy quantization. The energy levels are shown in fig. [Chart Description] Type: Energy level diagram. Main Elements: - Vertical Y-axis labeled "Energy" with an upward arrow indicating increasing energy. An infinity symbol (∞) is at the top of the axis. - Horizontal lines representing energy levels. - From bottom to top, these levels are labeled: - E1 - 4 E1 - 9 E1 - 16 E1 - 25 E1 - The levels are represented as horizontal lines within a vertical structure resembling a potential well with walls shown on the left and right, extending to infinity. The above equation indicates that a particle confined in a potential well can have only certain values of energy. Other energy values are not allowed. In other words restricting a micro particle to a certain region leads to the quantization of energy. These allowed values of energies for a particle for which Schrodinger equation can be solved is known as Energy Eigen values of the particle. The lowest energy allowed for the particle is E1 = h^2 / (8mL^2). It can not possess energy less than this. The energy E1 is called zero point energy. The zero point energy is the consequence of the uncertainty principle. If energy of the particle is zero, its momentum would be zero, and the uncertainty principle requires that the wavelength be infinite. Hence, in this case the particle cannot be confined to the box. Eqn (6) is called energy eigen values of nth particle. In this eq n is called quantum number which have values 1,2,3---------- Quantum mechanically. The zero point energy is the consequence of the uncertaintyprinciple. If E = 0 => p = 0 by relation E = p^2/2m , If p = 0 => lambda = infinity If lambda = infinity => The particle cannot be confined to a box. So, the particle must have certain minimum amount of kinetic energy. Classically, the particles may have zero point energy but Quantum mechanically it is not possible . To find wave function: The wave function corresponding to the allowed energy levels can be obtained as follows. ψ = A sin kx + B cos kx ∴ B = 0, k = nπ/L ⇒ ψ = A sin (nπx/L) |ψ|^2 = |A sin (nπx/L)|^2 But according to normalization condition, ∫ |ψ|^2 dx = 1 ∫[from 0 to L] |A sin (nπx/L)|^2 dx = 1 ∫[from 0 to L] A^2 sin^2 (nπx/L) dx = 1 A^2 ∫[from 0 to L] (1 - cos (2nπx/L))/2 dx = 1 A^2/2 [ ∫[from 0 to L] dx - ∫[from 0 to L] cos (2nπx/L) dx ] = 1 A^2/2 [ x - (sin (2nπx/L))/(2nπ/L) ] [from 0 to L] = 1 ⇒ A^2/2 [ (L - 0) - (sin(2nπ)L/L)/(2nπ/L) - (sin(0))/(2nπ/L) ] = 1 ⇒ A^2/2 [ (L - 0) - (0)/(2nπ/L) - 0 ] = 1 (Since sin(2nπ) = 0 and sin(0) = 0) ⇒ A^2/2 [ L - 0 ] = 1 ⇒ A^2/2 L = 1 ⇒ A^2 = 2/L ⇒ A = √(2/L) This is required wave function. Probability is given by $|\Psi|^{2} = \frac{2}{L} \sin^{2} (\frac{n\pi x}{L})$ Chart Description: * **Type:** Two sets of line charts, arranged vertically for different states (n=1, 2, 3). * **Arrangement:** Two columns of charts. The left column plots ψ(x) vs. Position, and the right column plots |ψ(x)|² vs. Position. * **Axes:** * Horizontal Axis (common to all charts): Labeled "Position" with range from 0 to L. * Vertical Axis: Labeled ψ₁, ψ₂, ψ₃ for the left column charts, and |ψ₁|², |ψ₂|², |ψ₃|² for the right column charts. These labels indicate the quantum number n = 1, 2, 3 for each horizontal level of plots. * **Content (Left Column - ψ(x)):** * ψ₁: A single positive lobe shaped like half a sine wave, from 0 to L. Starts at 0, peaks around L/2, ends at 0. * ψ₂: A sine wave shape with two lobes, one positive and one negative. Starts at 0, goes up to a peak, crosses the axis at L/2, goes down to a trough, ends at 0. * ψ₃: A sine wave shape with three lobes, positive, negative, and positive. Starts at 0, goes up to a peak, crosses the axis, goes down to a trough, crosses the axis again, goes up to a peak, ends at 0. * **Content (Right Column - |ψ(x)|²):** * |ψ₁|²: A single positive lobe shaped like a squared sine wave, from 0 to L. Starts at 0, peaks around L/2, ends at 0. Always positive. * |ψ₂|²: Two positive lobes, separated by zero at L/2. Starts at 0, peaks, goes down to 0 at L/2, peaks again, ends at 0. Always positive. * |ψ₃|²: Three positive lobes, separated by zeros. Starts at 0, peaks, goes down to 0, peaks, goes down to 0, peaks, ends at 0. Always positive. * **Figure Caption:** Fig. 4: Diagrams of ψ(x) & |ψn|² Other Relevant Text: 1) The allowed wave function $\Psi_{n}$ and the corresponding probability densities $|\Psi_{n}|^{2}$ for an electron trapped in a one dimensional potential well are plotted in Fig 4. While $\Psi_{n}$ may be negative, $|\Psi_{n}|^{2}$ is always positive. $|\Psi_{n}|^{2}$ gives the probability of finding the electron at a certain place within the well. 2) For n=1, $|\Psi_{1}|^{2}$ gives the maximum probability of locating the particle at the centre of the well i.e L/2 and decreases on either side of it. 3) However For n=2, $|\Psi_{2}|^{2}$ gives the maximum probability of locating the particle which is found at L/4 & 3L/4. For n=2, the probability of locating the particle at centre is zero. For higher values of 'n' the probability of locating the particle lies at several points in the close vicinity of each other and can be treated as continuum. Classically particle can have all possible valuesof probability. It simply means that particle can be found at all the places with equal probabilities. However quantum mechanically, the probability varies with the values of n. Dependence of quantization on width of well Or Using the relation E_n = (n^2 * h^2) / (8 * m * L^2), Where symbols have their usual meanings, show that the quantization of energy assumes great importance in atomic world. Let E_n and E_{n+1} are two adjacent energy levels ΔE = E_{n+1} - E_n ΔE = ((n + 1)^2 * h^2) / (8 * m * L^2) - (n^2 * h^2) / (8 * m * L^2) ΔE = (h^2 / (8 * m * L^2)) * ((n + 1)^2 - n^2) ΔE = (h^2 / (8 * m * L^2)) * (n^2 + 2n + 1 - n^2) ΔE = (h^2 * (2n + 1)) / (8 * m * L^2) ΔE = (h^2 * (2n)) / (8 * m * L^2) (∴ 2n >> 1; 1 can be neglected) ∴ ΔE = (h^2 * n) / (4 * m * L^2) Now assume two values for L • L = 1cm = 10⁻²m ΔE = ((6.63 × 10⁻³⁴)² * 2n) / (4 × 9.11 × 10⁻³¹ * (10⁻²)²) J ΔE = ((6.63 × 10⁻³⁴)² * 2n) / (4 × 9.11 × 10⁻³¹ * 10⁻⁴) J ΔE = ((6.63 × 10⁻³⁴)² * 2n) / (36.44 × 10⁻³⁵) J ΔE ≈ (43.96 × 10⁻⁶⁸ * 2n) / (36.44 × 10⁻³⁵) J ΔE ≈ (87.92 / 36.44) * n * 10⁻⁶⁸ * 10³⁵ J ΔE ≈ 2.41 * n * 10⁻³³ J The image calculation seems slightly different: ΔE = (1.2 × 10⁻³³ × n) J / (1.602 × 10⁻¹⁹) eV/J = n × 0.74 × 10⁻¹⁴ eV ΔE = n × 10⁻¹⁴ eV ----------(1) This energy difference between successive levels is so insignificant that an electron possess thermal K.E. = 10⁻³ eV. Then electron can easily move from one level to another without any external energy. Thus the energy levels are quasi continuous. **Chart Description:** **Diagram 1:** * Type: Energy level diagram. * Main Elements: * A rectangular box representing a potential well with boundaries at 0 and L. * Several closely spaced horizontal lines inside the box, representing energy levels. * Labels '0' and 'L' at the bottom x-axis. * Annotation 'L ≈ 1 cm' below the box. * A double-headed arrow pointing between two adjacent energy levels, labeled 'ΔE << KT'. **Diagram 2:** * Type: Energy level diagram. * Main Elements: * A rectangular box representing a potential well with boundaries at 0 and L. * Fewer, widely spaced horizontal lines inside the box, representing energy levels. * Labels 'E_n' and 'E_n+1' next to two adjacent energy levels on the left. * Labels '0' and 'L' at the bottom x-axis. * Annotation 'L ≈ 10 Å' below the box. * A double-headed arrow pointing between two adjacent energy levels, labeled 'ΔE >> KT'. **Textual Information:** * If L = 10Å = 10 x 10⁻¹⁰ m * ΔE = 0.74 x n eV * ΔE = (6.63×10⁻³⁴)²n² / (4×9.11×10⁻³¹(10⁻⁹)²) J * ΔE = (1.2 x 10⁻¹⁹ x n) / (1.602 x 10⁻¹⁹) = n x 0.74 eV ---------(2) * This energy difference between the adjacent levels is very large and electron cannot move into higher energy level without any external energy. i.e energy levels are far wider. * From eqn.(1) and (2),it is clear that of quantization energy depend on width of well 'L'. **Question Stem:** Que. Explain in short the phenomenon of tunneling that occurs when a beam of particles are incident on a potential barrier of finite width. OR Explain the (quantum mechanical ) barrier tunneling of electron on the basis of quantum mechanics. **Explanatory Text:** When an electromagnetic wave strikes at the interface of the two media, it is partly reflected and partly transmitted through the interface and enters the second medium. Similarly, the wave nature of micro particles makes it possible to get partly reflected from the boundary of the potential barrier and partly penetrate through the barrier. The penetration of a barrier by a quantum particle is called tunneling. A potential barrier is the opposite of potential well; it is a potential energy function with a maximum. Let us consider a potential barrier of height h and thickness L. The potential energy is zero for x<0 and x>L and potential energy has a value V for 0 < x < L. An electron of total energy E approaches the barrier from the left. From the view point of classical physics, the electron would be reflected from the barrier because its energy E is less than V. For the particle to overcome the potential barrier, it must have an energy equal to or greater than V. Quantum mechanics leads to entirely new result. It shows that there is a finite chance for the electron to leak to the other side of the barrier. We say that the electron tunneled through the potential barrier and hence in quantum mechanics, the phenomenon is called Quantum tunneling. **Diagram Description:** * **Type:** Potential energy diagram illustrating particle tunneling through a rectangular potential barrier. * **Coordinate Axes:** * Horizontal axis: Labeled Z, represents position. Origin is at 0, with a point labeled L to the right of 0. Arrow indicates positive Z direction. * Vertical axis: Labeled E, represents energy. Arrow indicates increasing energy upwards. * **Potential Barrier:** * Rectangular shape, extending vertically from the Z-axis. * Located along the Z-axis from Z=0 to Z=L. * Base is on the Z-axis between 0 and L. * Height is constant between Z=0 and Z=L. Labeled E₀ inside the barrier region. This corresponds to the potential energy V described in the text. * **Energy Level:** A horizontal dashed line labeled E, located below the top of the potential barrier (E < E₀/V). Represents the total energy of the incident particle. * **Regions:** * Region (I): Z < 0, to the left of the barrier. Potential energy is 0. Labeled "(I)" below the Z-axis. * Region (II): 0 < Z < L, within the barrier. Potential energy is E₀ (or V). Labeled "(II)" below the Z-axis. * Region (III): Z > L, to the right of the barrier. Potential energy is 0. Labeled "(III)" below the Z-axis. * **Wave Functions:** * Region (I): Shows a wave function ψ₁ which is a superposition of an incident wave (traveling right) and a reflected wave (traveling left). Labeled "Incident particle" with a right arrow and "Reflected particle" with a left arrow. ψ₁ is shown as a sinusoidal wave. * Region (II): Shows a wave function ψ₂ inside the barrier. It is depicted as an exponentially decaying wave, characteristic of a particle in a classically forbidden region (E < V). * Region (III): Shows a wave function ψ₃ to the right of the barrier. It is depicted as a sinusoidal wave with a smaller amplitude than the incident wave in Region (I). Labeled "Transmitted particle" with a right arrow. * **Labels and Annotations:** ψ₁, ψ₂, ψ₃, E, E₀, 0, L, (I), (II), (III), Incident particle, Reflected particle, Transmitted particle. **Final Text:** Let us consider a particle of energy E approaching the barrier from the left. The region around the barrier can be divided into three regions as shown in fig. If the particle is to go into region III, it must have energy equal to or greater than height h of the barrier. But according to quantum mechanics, particle can go into all the three regions regardless of its energy. The approximate transmission probability is given by Mathematical Formula: T = e⁻²ᵏ₂ᴸ Applications of the tunneling: 1) Tunnel diode 2) Scanning tunneling electron microscope(STM) 3) α - decay Q.10. Draw the sketches of Wave function and probability densities of a particle for first three states for finite potential well and discuss. Ans. Chart Description: The image displays two plots side-by-side, illustrating wave functions and probability densities for a particle in a finite potential well for the first three states. Left Chart: (a) Wave functions * Type: Line chart showing three different curves on a single set of axes. * X-axis: Labeled 'x', ranges from x=0 to x=L. * Y-axis: Represents the wave function, labeled with symbols Ψ₁, Ψ₂, and Ψ₃ for the three different states. There are no numerical scales shown. * Curves: * Ψ₁: A single hump-like curve, starting near zero at x=0, rising to a maximum, and decreasing towards zero near x=L. The curve extends slightly beyond x=0 and x=L before approaching zero. * Ψ₂: A curve with one full oscillation, starting near zero at x=0, going negative, crossing the x-axis, becoming positive, reaching a peak, crossing the x-axis again, becoming negative, reaching a trough, and ending near zero at x=L. The curve extends slightly beyond x=0 and x=L before approaching zero. * Ψ₃: A curve with two full oscillations, starting near zero at x=0, going positive, crossing the x-axis, becoming negative, crossing again, becoming positive, reaching a peak, crossing again, becoming negative, and ending near zero at x=L. The curve extends slightly beyond x=0 and x=L before approaching zero. * Labels: The three curves are labeled Ψ₁, Ψ₂, and Ψ₃ on the left side of the Y-axis, indicating increasing energy levels (or quantum numbers). Right Chart: (b) Probability densities inside non - rigid box * Type: Line chart showing three different curves on a single set of axes. This plot represents the square of the wave function magnitude, |Ψ|². * X-axis: Labeled 'x', ranges from x=0 to x=L. * Y-axis: Represents the probability density, labeled with symbols |Ψ₁|², |Ψ₂|², and |Ψ₃|² for the three different states. There are no numerical scales shown. All values are non-negative. * Curves: * |Ψ₁|²: A single hump-like curve, starting near zero at x=0, rising to a maximum, and decreasing towards zero near x=L. The curve extends slightly beyond x=0 and x=L before approaching zero. * |Ψ₂|²: A curve with two humps (positive regions), starting near zero at x=0, rising to a peak, decreasing to zero, rising to another peak, and decreasing towards zero near x=L. The curve extends slightly beyond x=0 and x=L before approaching zero. * |Ψ₃|²: A curve with three humps (positive regions), starting near zero at x=0, rising to a peak, decreasing to zero, rising to a peak, decreasing to zero, rising to a third peak, and decreasing towards zero near x=L. The curve extends slightly beyond x=0 and x=L before approaching zero. * Labels: The three curves are labeled |Ψ₁|², |Ψ₂|², and |Ψ₃|² on the left side of the Y-axis, corresponding to the respective wave functions. General description for both charts: Both charts share the same x-axis range (0 to L) and are vertically stacked to show the progression of states. The wave functions and probability densities are shown extending slightly beyond the boundaries x=0 and x=L before decaying to zero. Text below charts: The first three wave functions and probability densities when plotted against x are shown in the figure. The eigen functions are similar in appearance to those of infinite well except that they extend a little outside the box. Even though the particle energy E is less than the P.E. V₀ outside the box there is a definite probability that the particle is found outside the box. The particle energy is not enough to break through the walls of the box but it can penetrate the walls and leak out. This shows penetration of the particle into the classically forbidden region. Particle in a non-rigid box (Finite Potential Well) * Consider a particle of mass m moving with velocity v along the x-direction between x = 0 and x = L. * The walls of the box are not rigid. Hence it is represented by a potential well of finite depth. Step I : Let E be the total energy of the particle inside the box and V be its P.E. which is assumed to be zero within the box. The potential outside the box is finite say v₀ and v₀>E. The variation of potential with x is shown in fig 1.2 ```text Classically, the particle with energy E, Vo cannot be present in regions 1 and 2 outside the box. Consider the quantum mechanical picture of the particle in one dimensions. If Ψ is the wave function associated with the particle then schroedinger's time independent equation for it is, d²Ψ/dx² + 2m/ħ² (E – V) Ψ = 0 Step II : Consider the three regions 1,2,3 separately and let Ψ₁, Ψ₂, Ψ₃ be the wave functions in them separately. We have for region 1, d²Ψ₁/dx² + 2m/ħ² (E – V₀) Ψ₁ = 0 We have for region 2, d²Ψ₂/dx² + 2mE/ħ² Ψ₂ = 0 And for region 3, d²Ψ₃/dx² + 2m/ħ² (E – V₀) Ψ₃ = 0 Let 2mE/ħ² = k² And 2m(E – V₀)/ħ² = -k'² (as E < V₀) Then the equation in the three regions can be written as, d²Ψ₁/dx² - k'²Ψ₁ = 0 d²Ψ₂/dx² + k²Ψ₂ = 0 d²Ψ₃/dx² - k'²Ψ₃ = 0 Step III : The solution of these equation are of the form. Ψ₁ = A e^(k'x) + B e^(-k'x) for x<0 Ψ₂ = P . e^(ikx) + Q . e^(-ikx) for 0L Step IV : As x → ±∞, Ψ should not become infinite. Hence B = 0 and C = 0 **Chart Description:** * **Type:** Schematic diagram of a potential well. * **Main Elements:** * **Coordinate Axis:** A horizontal axis labeled 'x' and a vertical axis representing potential 'V'. * **Potential Profile:** The potential V is shown as a function of x. It is at a high value V₀ for x < 0 (Region I) and x > L (Region III), and at a low value (implied to be 0) for 0 < x < L (Region II). The potential profile is represented by vertical lines at x=0 and x=L, connected by horizontal lines at V=V₀ and V=0. * **Regions:** The x-axis is divided into three regions: I (x<0), II (0L). * **Energy Level:** A horizontal dashed line labeled 'E' is shown within Region II, indicating the energy of the particle. This line is below the potential level V₀ in regions I and III. * **Boundaries:** Vertical lines mark the boundaries at x = 0 and x = L. * **Labels:** Regions are labeled I, II, III. Potential values are indicated as V and V₀. The energy level is labeled E. The boundaries are labeled x=0 and x=L. * **Title:** Fig. 1.27.1 : Finite potential well ``` Hence the wave function in three region are $\Psi_I = A e^{k'x}$ $\Psi_{II} = P.e^{ikx} + Q.e^{-ikx}$ $\Psi_{III} = D.e^{-k'x}$ **Step V:** The constant A, P, Q and D can be determined by applying the boundary conditions. The wave function $\Psi$ and its derivative $\frac{d\Psi}{dx}$ Should be continuous in the region where $\Psi$ is defined. $\Psi_I (0) = \Psi_{II}(0)$ $\frac{d\Psi_I}{dx} |_{x=0} = \frac{d\Psi_{II}}{dx} |_{x=0}$ $\Psi_{II}(L) = \Psi_{III}$ $\frac{d\Psi_{II}}{dx} |_{x=L} = \frac{d\Psi_{III}}{dx} |_{x=L}$ **Chart/Diagram Description:** The image contains two plots side-by-side, labeled (a) and (b). Both plots are 2D line charts with a horizontal X-axis labeled 'x' and extending from x=0 to x=L, and a vertical Y-axis. Both charts show multiple curves, representing different energy levels. The region between x=0 and x=L is indicated on the X-axis. **(a) Wave functions:** Type: Line chart. X-axis: Labeled 'x', marked at 0 and L. Y-axis: Labeled $\Psi$. Content: Shows three different wave functions ($\Psi$) plotted against x. Each curve is a wave-like shape within the 0 to L region and decays rapidly outside this region. The curves are labeled $\Psi_1$, $\Psi_2$, and $\Psi_3$ from bottom to top on the Y-axis. **(b) Probability densities inside non - rigid box:** Type: Line chart. X-axis: Labeled 'x', marked at 0 and L. Y-axis: Labeled $|\Psi|^2$. Content: Shows the probability densities ($|\Psi|^2$) corresponding to the wave functions in (a), plotted against x. These curves are also wave-like within the 0 to L region but represent the square of the amplitude (always positive). They also decay rapidly outside the 0 to L region. The curves are labeled $|\Psi_1|^2$, $|\Psi_2|^2$, and $|\Psi_3|^2$ from bottom to top on the Y-axis. **Other Relevant Text:** * Using these four conditions, we get four equation from which the four constant A, B, C D can be determined. Thus the wave function can be known completely. * The first three wave function and probability densities when plotted against x are shown in fig 1 * The eigen function are similar in appearance to those of infinite well except that they extend a little outside the box. * Even though the particle energy E is less than the P.E. $V_0$ outside the box there is a definite probability that the particle is found outside the box. * The particle energy is not enough to break through the walls of the box but it can penetrate the walls and leak out. This shows penetration of the particle into the classically forbidden region. * The energy levels of the particle are still discrete but there a finite number of them. Such a limit exists because, soon the particle energy becomes equal to $V_0$. * For energies higher than this the Particle energy is not quantised but may have any value above $V_0$. * These predictions are unique in quantum mechanics and shows different behaviour from that expected in classical physics. UNIT-II Introduction to Quantum Computing [CO1,PO1,PO2 & PO12)] Complex Numbers: Quantum mechanics is full of complex numbers, numbers involving i = $\sqrt{-1}$. This section summarizes their most important properties. First, any complex number c can always be written in the form c = a + ib where both a and b are ordinary real numbers, not involving $\sqrt{-1}$. The number a is called the real part of c and b the imaginary part. Complex numbers can be manipulated pretty much in the same way as ordinary numbers can. A relation to remember is : $\frac{1}{i} = -i$ which can be verified by multiplying top and bottom of the fraction by i and noting that by definition i$^2$ = -1 in the bottom. The complex conjugate of a complex number c, denoted by c*, is found by replacing i everywhere by -i. In particular, if c = a + ib, where a and b are real numbers, the complex conjugate is c* = a - ib. The magnitude, or absolute value, of a complex number c, denoted by |c|, is the positive square root of c times its complex conjugate : $|c| = \sqrt{c^*c}$ If c = a + ib, where a and b are real numbers, multiplying out c*c shows the magnitude of c to be $|c| = \sqrt{a^2 + b^2}$ Complex numbers of magnitude one can always be written in the form $e^{i\alpha}$ where $\alpha$ is an ordinary real number. The critically important Euler identity says that: $e^{i\alpha} = \cos(\alpha) + i \sin(\alpha)$ Note that the only two real numbers of magnitude one, 1 and -1, are included for $\alpha=0$, respectively $\alpha=\pi$. Functions as vectors: The second mathematical idea that is critical for quantum mechanics is that functions can be treated in a way that is fundamentally not that much different from vectors. **Text Extraction:** A vector $\vec{f}$ (which might be velocity $\vec{v}$, linear momentum $\vec{p} = m\vec{v}$, force $\vec{F}$, or whatever) is usually shown in physics in the form of an arrow: Figure 1: The classical picture of a vector. However, the same vector may instead be represented as a spike diagram, by plotting the value of the components versus the component index: Figure 2: Spike diagram of a vector. (The symbol i for the component index is not to be confused with $i = \sqrt{-1}$) In the same way as in two dimensions, a vector in three dimensions, or, for that matter, in thirty dimensions, can be represented by a spike diagram: Figure 3: More dimensions. For a large number of dimensions, and in particular in the limit of infinitely many dimensions, the large values of i can be rescaled into a continuous coordinate, call it $\mathbf{x}$. For example, $\mathbf{x}$ might be defined as i divided by the number of dimensions. In any case, the spike diagram becomes a function $f(\mathbf{x})$: **Chart/Diagram Description:** **Figure 1:** * **Type:** Cartesian Coordinate Diagram with a vector. * **Main Elements:** * **Coordinate Axes:** X-axis (labeled 'x') and Y-axis (labeled 'y'). They appear to intersect at the origin (0,0). * **Point:** A blue filled circle labeled 'm' is located near the origin. * **Vector:** An arrow labeled $\vec{f}$ originates from the point 'm' and points towards the upper right. * **Labels:** Vertical arrow labeled $f_y$ pointing upwards from the origin along the y-direction. Horizontal arrow labeled $f_x$ pointing rightwards from the origin along the x-direction. These labels seem to represent the components of the vector $\vec{f}$. * **Title:** Figure 1: The classical picture of a vector. **Figure 2:** * **Type:** Spike Diagram. * **Main Elements:** * **Coordinate Axes:** Vertical axis labeled $f_i$. Horizontal axis labeled $i$ with tick marks at 1 and 2. * **Spikes:** Two vertical arrows originating from the horizontal axis. The spike at index 1 is labeled $f_x$ and points upwards. The spike at index 2 is labeled $f_y$ and points upwards. The height of the spike at index 1 is labeled $f_x$ and the height of the spike at index 2 is labeled $f_y$. * **Title:** Figure 2: Spike diagram of a vector. **Figure 3:** * **Type:** Two separate Spike Diagrams shown side-by-side. * **Main Elements (Left Diagram):** * **Coordinate Axes:** Vertical axis labeled $f_i$. Horizontal axis labeled $i$ with tick marks at 1, 2, and 3. * **Spikes:** Three vertical arrows originating from the horizontal axis. The spike at index 1 is labeled $f_x$. The spike at index 2 is labeled $f_y$. The spike at index 3 is labeled $f_z$. All spikes point upwards. The height of the spikes are labeled $f_x$, $f_y$, and $f_z$ respectively. * **Main Elements (Right Diagram):** * **Type:** Spike Diagram representing many dimensions. * **Coordinate Axes:** Vertical axis labeled $f_i$. Horizontal axis labeled $i$ with tick marks at 1 and 30. * **Spikes:** Multiple vertical arrows originating from the horizontal axis, densely packed between indices 1 and 30. The tops of the spikes form a curve that rises from the left, dips in the middle, and rises again towards the right, resembling a wave. Each spike has a small asterisk or dot at its top. * **Title:** Figure 3: More dimensions. **Chart Description:** * **Type:** 2D Graph. * **Main Elements:** * **Coordinate Axes:** Y-axis labeled `f(x)`, X-axis labeled `x`. * **Curve:** A curved line representing a function `f(x)` in the upper part of the graph. * **Shaded Area:** The region below the curve and above the x-axis is shaded, extending between two vertical lines (implicitly marking a range on the x-axis). **Textual Information:** Figure 4: Infinite dimensions. In this way, a function is just a vector in infinitely many dimensions. For example, e^x s an eigen function of the operator d/dx with eigenvalue 1, since d/dx (e^x) = 1e^x However, eigen functions like e^x are not very common in quantum mechanics since they become very large at large X and that typically does not describe physical situations. The eigen functions of d/dx that do appear a lot are of the form e^ikx, where i = sqrt(-1) and k is an arbitrary real number. The eigenvalue is ik : d/dx e^ikx = ike^ikx Function e^ikx does not blow up at large X ; in particular, the Euler identity says: e^ikx = cos(kx) + i sin(kx) Concept of Operator: The wave function ψ(x) which when operated by an operator O are merely multiplied by some constant 'k' if Ôψ(x) = kψ(x) Then wave function ψ(x) are called Eigen function of the operator O and the various possible values k are called eigen values of operator O. For example, We consider the function- ψ(x) = Sin4x When the wavefunction is operated by operator (-d^2/dx^2) The result is Ôψ(x) = kψ(x) (-d^2/dx^2)(Sin4x) = 16(Sinn4x) Thus Sin4x is Eigen function and 16 is Eigen value of the operator (-d^2/dx^2) Eigen Value & Eigen Function: **Textual Information:** **Introduction:** Eigenvalues are associated with eigenvectors in Linear algebra. Both terms are used in the analysis of linear transformations. Eigenvalues are the special set of scalar values that is associated with the set of linear equations most probably in the matrix equations. The eigenvectors are also termed as characteristic roots. It is a non-zero vector that can be changed at most by its scalar factor after the application of linear transformations. And the corresponding factor which scales the eigenvectors is called an eigenvalue. **Eigenvalue Definition:** Eigenvalues are the special set of scalars associated with the system of linear equations. It is mostly used in matrix equations. 'Eigen' is a German word that means 'proper' or 'characteristic'. Therefore, the term eigenvalue can be termed as characteristic value, characteristic root, proper values or latent roots as well. In simple words, the eigenvalue is a scalar that is used to transform the eigenvector. The basic equation is Ax = λx The number or scalar value "λ" is an eigenvalue of A. **Eigenvalues as characteristic or latent roots:** Eigenvalues are also known as characteristic or latent roots, is a special set of scalars associated with the system of linear equations. In Mathematics, an eigenvector corresponds to the real non zero eigenvalues which point in the direction stretched by the transformation whereas eigenvalue is considered as a factor by which it is stretched. In case, if the eigenvalue is negative, the direction of the transformation is negative. For every real matrix, there is an eigenvalue. Sometimes it might be complex. The existence of the eigenvalue for the complex matrices is equal to the fundamental theorem of algebra. **What are Eigen Vectors?** Eigenvectors are the vectors (non-zero) that do not change the direction when any linear transformation is applied. It changes by only a scalar factor. In a brief, we can say, if A is a linear transformation from a vector space V and x is a vector in V, which is not a zero vector, then v is an eigenvector of A if A(X) is a scalar multiple of x. An Eigenspace of vector x consists of a set of all eigenvectors with the equivalent eigenvalue collectively with the zero vector. Though, the zero vector is not an eigenvector. Let us say A is an "n × n" matrix and λ is an eigenvalue of matrix A, then x, a non-zero vector, is called as eigenvector if it satisfies the given expression; Ax = λx x is an eigenvector of A corresponding to eigenvalue, λ. **Mathematical Formulas/Equations:** Ax = λx (Appears multiple times) A(X) is a scalar multiple of x (Conceptual expression) **Chart/Diagram Description:** * **Type:** Diagram illustrating a mathematical relationship (vector transformation). * **Main Elements:** * Labels: Matrix, Eigenvector, Eigenvalue. * Vectors/Arrows: * An arrow labeled 'v' originates near "Eigenvector" and points upwards and slightly right. * An arrow labeled 'Av' originates near "Matrix" and points upwards and slightly left. * An arrow labeled 'λv' originates near "Eigenvalue" and points upwards and slightly left, parallel to 'Av'. * Relative Position and Direction: The diagram shows three sources ("Matrix", "Eigenvector", "Eigenvalue") influencing two resulting vectors ('Av' and 'λv'). The vectors 'Av' and 'λv' are parallel and appear to point to the same conceptual resulting vector, indicating their equality (Av = λv, a specific case of Ax = λx where x=v). The arrows originating from "Matrix" and "Eigenvalue" indicate transformations or scaling being applied to "Eigenvector". The classical bit The bit that we all are familiar with, is the most basic unit of information. It either represents a 1 or a 0. We can think of it as analogous to a switch being on or off. A bit's current status, either 1 or 0, is known as its state. The quantum bit The quantum bit can represent 0, 1, or both simultaneously. This seems quite counter-intuitive. What does it mean to be two different things simultaneously? This phenomenon is called superposition. We will come back to this in a future lesson about quantum mechanics. For now, we just need to accept that it's possible to be 0 and 1 at once. Quantum states The first take away here is that the superposition of 0 and 1 is not some third possible state of a bit. It is a special state that we cannot describe by using a classical bit. Let's see why it is so special. Notation Let's introduce some notations to distinguish quantum states from classical states. We will represent the 0 state in quantum as "|0>" and the 1 state as "|1>". Superposition state Think of a slider moving between the values from 0 to 1. A superposition of 0 and 1 means our current state is somewhere on the slider, it could be a little more towards the 0 side, or it could be a little more towards the 1 side. We see from this slider analogy that there are infinitely many possible states that we can call "being both 0 and 1 at the same time". We can think of such a quantum state as a whole, with some part being a 0 and the remaining part of it being a 1. Let's try to capture this concept using our new notation. We can write a superposition state as a combination of |0> and |1> in the form of |0>+|1>. An equal superposition state would mean 50% of the qubit is a 0 and the other 50% of it is a 1. Using this way, we can think of the |0> state to be 100% 0 and 0% |1>. Similarly, we can say the |1> state is 0% 0 and 100% 1. Differences between Qubits vs Bits: * When we consider bit in traditional computing technology, bits refer only to the binary values such as 0s and 1s, and they cannot be considered for other values. Whereas in qubits, it represents 0s, 1s, and a superposition of both the values. That means it can be used to represent the combination of 0s and 1s in quantum computing, where it is much important to notify all the values in the system. * When bit storing the information of binary digits, qubits store the combination of binary digits, which helps the qubits in quantum computing work three times as fast as a conventional computer system. The information stored and the data transfer is huge, which helps to transfer the information faster. * When the problem is to be solved on the computer, bits approach the problem as if in a hit and trial run. This is due to the fact that one value is considered at a time, and parallel processing is not happening when the problem has to be solved. When the same problem has to be solved using quantum computing, it is approached with parallel processing by supporting all four values at a time and solving it at a faster pace. Quantum computers: * When more qubits are added to the quantum computer, the power to do the processing increases at an exponential rate. In contrast, when bits are added to the normal computer, the power will not increase, and the operations will be done at the same pace as one at a time. In quantum computing, this happens due to superposition. * It is extremely difficult to build quantum computers because they need extreme isolation and quantum objects' proper temperature. This is not the case with traditional computers, which anyone with hardware knowledge can build and make it work for all the needed conditions for the user. Hence, the number of quantum computers is very less, and their use is recently being increased. * The storage space required by traditional computers for bits is huge, and it takes up lots of room. This can be avoided for qubits as huge information can be stored in the system with a small area. As the systems and devices are getting smaller, qubits help reimagine the technological world with really small size devices being handy to carry everywhere. * The scientific world can be viewed with different light with the help of qubits as it helps to modify and recalculate the physical phenomena, even though it is really huge, within a short span of time than the normal computers and make the process really easy for all who is beneficial with the same. Wave function: Wave represents the propogation of a disturbance in a medium. . Every wave is characterized by some quantity known as the wave variable which varies with space and time. eg. the sound waves have pressure as the wave variable which varies with space and time, the waves on strings have displacement y as the wave variable, the light waves consists of variations of electric and magnetic fields in space. In analogy with these waves known to us, it was suggested that the matter waves or de Broglie waves associated with them and it should be a function of space and time. **The wave variable associated with the matter waves is called the wave function Ψ(x,y,z,t) and it mathematically describes the motion of an electron.** The wave function - * Ψ is a function of both position (x,y,z) & time (t). * Ψ is a complex quantity containing real & imaginary terms. * Ψ has no direct physical significance, as * Ψ is not an observable quantity. Classically the intensity of a wave motion is proportional to the square of the amplitude of the wave. Thus the region of space where the particle is more likely to be found at time 't' are those where the intensity of the particle is more likely to be found at time t are those where intensity of the field |Ψ|² is large. i.e. Probability of finding the particle in an infinitesimal volume is proportional to |Ψ(x,y,z)|² dx dy dz at time t. P ∝ |Ψ(x,y,z)|² dv Physical significance. Q. Explain physical significance of wave function Ψ. Q. What is physical significance of wave function ψ. Q. Give physical significance of wave function ψ. Ans. The wave function ψ mathematically describes the motion of an electron (ie particle). But since ψ is not observable quantity therefore ψ has no direct physical significance. The wave function ψ is a complex quantity, but the probability must be real. Therefore to make it a real quantity, Max-Born German physicist showed that if ψ is multiplied by its complex conjugate ψ* then square of absolute valve of the wave function |Ψ|² is proportional to the probability of a particle being in unit volume of space. Thus the probability of finding a particle between x & x+dx, y & y+dy, z & z+dz is- P ∝ ∫∫∫ ψψ* dxdydz Where ψ* is a complex of ψ ∴ |ψ|² = ψψ* and dv = dxdydz P ∝ ∫_{x}^{x+dx} ∫_{y}^{y+dy} ∫_{z}^{z+dz} |ψ|² dv Normalization Condition Since the particle must be found somewhere in the space i.e in the universe, the total probability of finding the partied in entire space is unity. ∫_{-∞}^{+∞} ∫_{-∞}^{+∞} ∫_{-∞}^{+∞} |ψ|² dv = 1 This condition is known as the Normalization Condition and a wave function ψ satisfying above condition is said to be normalized. Whenever wave functions are normalized. if |Ψ|²dv equals to the probability that a particle will be found in volume dv. P = |Ψ(x,y,z)|²dv Q. State the properties of wave function "ψ". OR Conditions of well behaved wave function (2m) 1) The wave function ψ must be single valued function: Any physical quantity can have only one value at a point. For this reason the function related to a physical quantity cannot have more than one value at that point. If it has more than one value at a point it means that there is more than one value of probability of finding the particle at that point. 2) ψ Should be finite, at all point: The wave function ψ must be finite everywhere. Even if x → ∞ or - ∞, y → ∞ or - ∞, z → ∞ or - ∞, the wave function should not tend to infinity. It must remain finite for all values of x, y, z. If ψ is infinite, it would imply an infinitely large probability of finding the particle at that point. 3) ψ must be continuous: ψ function should be continuous across any boundary. Since ψ is related to a physical quantity. It cannot have a discontinuity at any point. Therefore, the wave function ψ and its space derivatives ∂ψ/∂x, ∂ψ/∂y & ∂ψ/∂z should be continuous across any boundary. Since ψ is related to a real particle, it cannot have discontinuity at any boundary where potential changes. **Chart Description:** * **Figure a:** * Type: Line chart. * Coordinate Axes: X-axis labeled 'x', Y-axis labeled '|ψ|²' with an upward arrow indicating direction. The origin is not explicitly labeled, but a point on the X-axis is labeled 'x₀'. * Data Series: A curved line showing a wave-like function oscillating and decreasing in amplitude as 'x' increases, with a dashed vertical line at x₀ crossing the curve. The curve is continuous. * **Figure b:** * Type: Step chart / Line chart with steps. * Coordinate Axes: X-axis labeled 'x', Y-axis labeled '|ψ|²' with an upward arrow indicating direction. A point on the X-axis is labeled 'x₀'. * Data Series: A horizontal line segment from some point on the left up to x₀, then a vertical step upwards at x₀, followed by another horizontal line segment to the right of x₀ at a higher '|ψ|²' value. There is a discontinuity (step) at x₀. 4) ψ must be normalized function: Since the particle must be found somewhere in the space i.e. in the universe, the total probability of finding the particle in entire space is unity. **Mathematical Formula:** ∫⁻₊ ∫⁻₊ ∫⁻₊ |ψ|² dv = 1 This condition is known as the Normalization Condition and a wave function ψ satisfying above condition is said to be normalized. Q. Write down Schrodinger's time independent wave equation for matter waves. Schrodinger's wave eq. (2 m) Schrodinger wave eq is of two type. 1) Time dependent equation.