UNIT I Quantum Physics Concepts Introduction to Vectors and Coordinate Systems Vectors in physics are quantities that possess both magnitude and direction crucial for describing various physical phenomena Common examples include velocity force and acceleration For instance velocity indicates how fast an object moves and in which direction while force describes the push or pull acting on an object also defined by its magnitude and direction Vectors are typically represented graphically as arrows with the length representing magnitude and the arrowhead indicating direction They can be added or subtracted using the parallelogram rule or triangle method Understanding vectors is essential for analyzing motion and interactions in physics Algebraic Form As ordered pairs in 2D or triplets in 3D e g v x y or v x y z Geometric Form As arrows in space Coordinate Systems Coordinate Systems provide a framework for locating points in space The most common types include Cartesian Coordinate System Uses perpendicular axes x y and z to define positions in space Each point is defined by its coordinates x y in 2D or x y z in 3D Polar Coordinate System Represents points in terms of distance from a reference point and an angle from a reference direction Common in 2D Cylindrical and Spherical Coordinate Systems Extensions for 3D where points are defined by radius angle and height cylindrical or radius and two angles spherical Understanding vectors and coordinate systems is essential in physics engineering and computer graphics as they form the basis for analyzing motion forces and spatial relationships Vector addition is the operation in which two vectors are combined to get their resultant vector which will produce same results as A and B together Suppose we take two vectors A and B then that can be added together using vector addition and the resultant vector is A B Vectors addition and subtraction Vector Addition Suppose we have two vectors Example 2 3 4 1 Vector Subtraction 2 4 3 1 6 2 Vector subtraction of two vectors A and B is represented by A B and it is nothing but adding the negative of vector B to the vector A i e A B A B Thus the subtraction of vectors involves the addition of vectors and the negative of a vector The result of vector subtraction is again a vector Example Using the same vectors 2 3 4 1 Laws of Vector Addition 2 4 3 1 2 4 There are three basic laws of vector addition that are used to add vectors and that include 1 Triangle Law of Vector Addition The triangle law of vector addition arranges the two vector and its resultant vector of addition in the form of a triangle In this triangle we have the third side of the triangle as resultant vector R and an angle between two vectors Formula for Magnitude of Resultant of any two vectors is given by R A2 B2 2ABcos where R is the Resultant of A and B A and B are two vectors is the angle between A and B Formula for the direction of resultant vector of A and B i e is given by where tan 1 B sin A B cos is the angle of the Vector from positive x axis A and B are two vectors is the angle between A and B 2 Parallelogram Law of Vector Addition It is a fundamental principle in vector mathematics that explains how to add two vectors It describes that when two vectors are added together their sum is represented by the diagonal of the parallelogram that starts from the same point as the two vectors which is called resultant vector Magnitude of Resultant Vector R R P2 Q2 2PQcos Where A and B are the magnitudes lengths of vectors P and Q respectively is the angle between vectors P and Q P2 and Q2 represent the squares of the magnitudes of vectors P and Q and cos is the cosine of the angle between the vectors Direction of Resultant Vector R Let the Resultant Vector R make angle with vector P then the direction of resultant vector is given as follows tan Q sin P Q cos Where is the angle between the resultant vector and vector P 3 Polygon Law of Vector Addition In the specific case of vector A vector B vector C and vector D the results can be obtained by drawing a polygon with the vectors as its sides and then taking the closing side of the polygon in the opposite direction The magnitude and direction of the resultant will be the same as the magnitude and direction of the closing side of the polygon On joining all vectors by connecting one s tail with the other s head without changing their magnitude and direction we get a Polygon and the vector joining the tail of the first and the head of the last vector is our Resultant vector R A B C D Introduction to Quantum Physics Quantum Physics is the study of the behavior of matter and energy at the molecular atomic nuclear and even smaller microscopic levels In the early 20th century scientists discovered that the laws governing macroscopic objects do not function the same in such small realms What Does Quantum Mean Quantum comes from the Latin meaning how much It refers to the discrete units of matter and energy that are predicted by and observed in quantum physics Differences between Classical and Quantum Mechanics Classical mechanics and quantum mechanics are two fundamental branches of physics that describe the behavior of physical systems at different scales S No Aspect Classical Mechanics Quantum Mechanics 1 2 Fundamental Concepts Determinism Newton s laws of motion and gravity Wave particle duality uncertainty principle Deterministic future states can be predicted with precision Probabilistic outcomes are described by probabilities 3 State Description State defined by specific values of position and momentum State defined by a wave function encompassing probabilities 4 Superposition No concept of superposition states are mutually exclusive 5 Measurement Measurements do not affect the system significantly Uncertainty Not inherent Superposition allows particles to be in multiple states simultaneously Measurement collapses the wave function affecting the state Heisenberg s uncertainty principle Scale of Applicability Applies to macroscopic objects and everyday phenomena Essential for atomic and subatomic scales microscopic scale Mathematical Framework Uses calculus vector algebra and differential equations Uses linear algebra operators and complex numbers Energy Energy is continuous and can take any value Energy levels are quantized only discrete values are allowed 10 Particles 11 Forces Particles follow definite paths and trajectories Particles exhibit both particle and wave characteristics Forces