Connecting Dirac Notation and Linear Algebra to Quantum Mechanics, Cheat Sheet of Quantum Mechanics

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(5) Significance in Quantum Mechanics The Gram-Schmidt process is essential in quantum mechanics for creating a valid basis from a given set of vectors. For example, it is used to converting a set of non-orthogonal wavefunctions into an orthonormal basis. Problem: Constructing an Orthonormal Basis for Quantum States Given two non-normalized state vectors in a Hilbert space: tm>=(]). 9 =(7) 1. Use the Gram-Schmidt orthonormalization procedure to obtain an orthonormal basis {|e,>.|e2>) from these vectors. Note: Use the complex inner product (rly) = 2'y, 2. Normalize the state [> = (24) and express il as a superposition ly = 1) +e [e2). Compute the measurement probabilities ||? and |e,|?. 3. Suppose an observable A has eigenvectors |¢,) and |e,) with eigenvalues +1 and -1, respectively. Find the expectation value Solution 1. Constructing the Basis (Gram-Schmidt) Normalize the first vector ley = ge = (1) Tull” Vek Compute the projection of |u,} onto |e,>? 1+? gel? = se u(t)= FT |e) = Ia) ~ éealu)ley) = (1) - Iny=(2)) eli) ~ oe (A) 36 Important Note on Phase Factors: . ee AD . In quantum mechanics, a global phase factor (like aw) has no physical meaning and does not change the physical state (the direction of the ray in Hilbert space). Definition: Ray in Quantum Mechanics A Ray in a complex Hilbert space is a one-dimensional complex subspace. All vectors lying on this same ray differ only by sealing (magnitude) or phase (direction in complex plane), but they represent the identical physical state. Thus, in quantum mechanics, 2 state is defined not by a single vector, but dy the ray itself, tere Ve ati) -1) > 4) Result: The orthonormal basis is {|¢,).les>}- Physical Insight: Global Phase Factor In the previous step, we obtained the orthogonal vector: . 1 i an per] la) 2 (4) (Jee (24) You might wonder: “Can we drop the complex coefficient? The answer is YES. + Reason: In quantum mechanics, a global phase factor (like ¢°'"'") has no physical meaning. It does not affect probabilities (/yI?) or expectation values. The state vector represents a "ray" in Hilbert space, not a single point. + Convention: We always choose the simplest form for our basis vectors. Therefore, we can discard the complex scalar and define the direction using anly the real vector part: ny» (4) 2. Superposition and Probabil Norm: fe the state [a> lvl = via +i+a? = V6, Normalized ly> = A 2 L+i 37 These notes are organized to help students from high school to junior-year university level build the structure through linear algebra and then move into wavefunctions, so that quantum mechanics can be learned in a clear and complete way. My main goal, and also the biggest strength of these notes, is to help students use the linear algebra they learned in mathematics more directly in studying quantum mechanics. Please see my Insta for more details: trose.quantum