gold 2/3/2026. Here are some simple snippets for numerical methods. The goal is to use Tcl's minimalism as a learning tool. Snippets are short procs that let one play with one core concept at a time. All snippets are Playground V9 safe. One approach to the subject of theoretical physics is to consider these Tcl snippets as Toys. Some snippets here are listed as Toys. These Tcl procs are tiny entry points into theoretical physics. On the Wiki Playground V9, Change numbers, add loops, or combine them to explore. Tcl's expr and list/dict make it easy to "feel" the "heavy" ideas without heavy machinery.
The quantum circuit tool only visualizes circuit structure, whereas no state vector simulation, probabilities, or actual measurement outcomes are derived. The example tool is not a full quantum emulator. Meaning, limited scope for tutorial purposes.
Advisor requests similar to previous snippets, but on topic of computing circuits ref Feynman. The Ideas Seemed to work, but maybe drawbacks?
The page presents Tcl code snippets demonstrating three fundamental quantum computing concepts: superposition, entanglement, and interference. These educational examples aim to make quantum mechanics accessible through minimalist programming on the Tcl Playground V9 platform. The following analysis examines how the code implements quantum principles, evaluates the floating-point precision observed in outputs, and suggests improvements for clarity and educational value.
The Hadamard gate creates quantum superposition by transforming a definite state into an equal probability mixture of basis states. The apply_hadamard procedure implements this transformation using the mathematical formula where each new amplitude equals the scaled sum or difference of input amplitudes. The scaling factor of one divided by the square root of two ensures probabilities sum to one, a fundamental requirement in quantum mechanics. When the procedure receives an input of 1.0 and 0.0 representing the ground state, the output produces two amplitudes of approximately 0.707, each corresponding to a probability of 0.5. This equal weighting demonstrates superposition where the quantum bit exists simultaneously in both zero and one states until measurement occurs.
The code tests three scenarios to illustrate different aspects of superposition behavior. Starting from the ground state produces equal positive amplitudes. Starting from the excited state yields one positive and one negative amplitude, demonstrating phase relationships. The third test begins with a superposition state already containing a minus sign between components, and the Hadamard operation collapses this back to a definite state with amplitude 1.0 for one outcome and 0.0 for the other. This reversibility property shows how Hadamard gates can both create and destroy superposition depending on the input state. The outputs from Playground V9 match expected values closely, with small discrepancies in the final digits arising from standard floating-point arithmetic limitations of expr function.
The create_bell procedure demonstrates quantum entanglement by generating a Bell state, one of four maximally entangled two-qubit states. This particular Bell state produces a superposition where both qubits are either zero together or one together, with no possibility of mismatched outcomes. The procedure returns four amplitudes representing the basis states 00, 01, 10, and 11, with only the first and last having non-zero values of approximately 0.707. Squaring these amplitudes gives probabilities of 0.5 for measuring both qubits in the same state, whether both zero or both one. The middle two amplitudes remain exactly zero, indicating measuring one qubit as zero and the other as one never occurs. This correlation pattern reveals the essence of quantum entanglement. Measuring the first qubit immediately determines the second qubit's state, even though neither qubit has a definite value before measurement. If measurement reveals the first qubit as zero, the second qubit must also be zero. If the first qubit measures as one, the second qubit must be one. This perfect correlation exists regardless of the physical separation between qubits, a phenomenon Einstein famously called "spooky action at a distance." The Tcl implementation correctly captures this behavior through the amplitude structure, though the code comments note this visualization tool does not simulate actual measurement outcomes or state vector evolution during gate operations.
The interfere procedure models quantum interference by combining amplitudes from two different paths to the same final state. The phase_sign parameter controls whether paths add constructively with a value of positive one or destructively with negative one. When two equal paths combine constructively, amplitudes add to produce approximately 1.414, and the probability reaches nearly 2.0, demonstrating constructive reinforcement. When the same paths combine with opposite phase, amplitudes cancel to zero, eliminating that outcome entirely. This destructive interference explains quantum phenomena like the double-slit experiment where certain detector positions show no particles despite multiple paths leading there.
The third test case uses unequal path amplitudes of 0.8 and 0.6 to demonstrate partial interference effects. Constructive combination yields a total amplitude of 1.4 and probability of 1.96, showing how unequal contributions still reinforce. The procedure correctly implements the quantum rule that amplitudes add before squaring to find probabilities, contrasting with classical probability where probabilities themselves add. This amplitude-first approach enables interference patterns impossible in classical physics. Real quantum circuits exploit interference by carefully adjusting phase relationships to amplify correct answer amplitudes while canceling incorrect ones, the foundation of quantum algorithms like Grover's search.
Richard Feynman was an inspiring theoretical physicist whose work on path integrals, quantum electrodynamics, and nanotechnology reshaped our understanding of quantum mechanics. In his 1981 lecture "Simulating Physics with Computers," he argued that classical computers struggle to simulate quantum systems efficiently, suggesting nature might compute quantum mechanically. David Deutsch built on this in 1985 by formalizing the quantum Turing machine and showing quantum computers could solve problems intractable classically, such as factoring via later Shor's algorithm. Their emphasis on elegant structures, superposition, entanglement, and interference offers rich conceptual inspiration for clean, concise code.
The printout values reflect roughly 17-digit precision output from a typical double-precision computation. It's not "true exact" beyond 5 significant figures. Extra significant figures are used to check the calculations from other computer set-ups, not necessarily to infer accuracy of data measurements here. Typically, the slight differences in decimal places on far right of decimal point are normal floating-point behavior in Tcl's expr.
Snippets Physics Concepts Quantum double
References
This is a draft, still debugging on Playground V9.
# Compatible with Tcl/Tk 8.6+
# Superposition: Hadamard creates equal amplitudes V2
# TCL source code follows
# Written for Windows 11 on ActiveState TCl
# Working on TCL Playground V9, strict ASCII only
# Optimized for collegiate IT lab environments
# Working under TCL version 8.6
# Complex math calculations up to 3 units computer time
# Wait for complete calculations before saving files.
# TCL club, 02/5/2026
# Superposition: Hadamard creates equal amplitudes (like exploring paths in parallel)
proc apply_hadamard {amp0 amp1} {
set scale [expr {1.0 / sqrt(2)}]
set new0 [expr {$scale * ($amp0 + $amp1)}]
set new1 [expr {$scale * ($amp0 - $amp1)}]
return [list $new0 $new1]
}
puts "Superposition Tests:"
set st [apply_hadamard 1.0 0.0]; puts "From |0>: amps $st probs [expr {[lindex $st 0]**2}] [expr {[lindex $st 1]**2}]"
set st [apply_hadamard 0.0 1.0]; puts "From |1>: amps $st probs [expr {[lindex $st 0]**2}] [expr {[lindex $st 1]**2}]"
set st [apply_hadamard [expr {1.0/sqrt(2)}] [expr {-1.0/sqrt(2)}]]
puts "From (|0>-|1>)/sqrt(2): amps $st probs [expr {[lindex $st 0]**2}] [expr {[lindex $st 1]**2}]"From |0>: amps ~0.707 0.707 probs ~0.5 0.5
From |1>: amps ~0.707 -0.707 probs ~0.5 0.5
From (|0>-|1>)/sqrt(2): amps 0.0 1.0 probs 0.0 1.0
(tcl) 3 % puts "Superposition Tests:"
Superposition Tests:
(tcl) 4 % set st [apply_hadamard 1.0 0.0]; puts "From |0>: amps $st probs [expr {[lindex $st 0]**2}] [expr {[lindex $st 1]**2}]"
From |0>: amps 0.7071067811865475 0.7071067811865475 probs 0.4999999999999999 0.4999999999999999
(tcl) 5 % set st [apply_hadamard 0.0 1.0]; puts "From |1>: amps $st probs [expr {[lindex $st 0]**2}] [expr {[lindex $st 1]**2}]"
From |1>: amps 0.7071067811865475 -0.7071067811865475 probs 0.4999999999999999 0.4999999999999999
(tcl) 6 % set st [apply_hadamard [expr {1.0/sqrt(2)}] [expr {-1.0/sqrt(2)}]]
0.0 0.9999999999999998
(tcl) 7 % puts "From (|0>-|1>)/sqrt(2): amps $st probs [expr {[lindex $st 0]**2}] [expr {[lindex $st 1]**2}]"
From (|0>-|1>)/sqrt(2): amps 0.0 0.9999999999999998 probs 0.0 0.9999999999999996# Entanglement: Bell state (H + CNOT) creates perfect correlations
proc create_bell {} {
set scale [expr {1.0 / sqrt(2)}]
# Result: (|00> + |11>)/sqrt(2) after H on qubit 0 then CNOT 0->1
return [list $scale 0.0 0.0 $scale]
}
puts "Entanglement Tests:"
set st [create_bell]; puts "Bell amps |00> [lindex $st 0] |01> [lindex $st 1] |10> [lindex $st 2] |11> [lindex $st 3]"
puts "Probs: |00>=[expr {[lindex $st 0]**2}] |11>=[expr {[lindex $st 3]**2}] others=0.0"
puts "Correlation example: measuring qubit 0 as 0 forces qubit 1 to 0; as 1 forces 1."Bell amps |00> 0.707 |01> 0.0 |10> 0.0 |11> 0.707
Probs: |00>=0.5 |11>=0.5 others=0.0
Correlation example: ...
(tcl) 3 % puts "Entanglement Tests:"
Entanglement Tests:
(tcl) 4 % set st [create_bell]; puts "Bell amps |00> [lindex $st 0] |01> [lindex $st 1] |10> [lindex $st 2] |11> [lindex $st 3]"
Bell amps |00> 0.7071067811865475 |01> 0.0 |10> 0.0 |11> 0.7071067811865475
(tcl) 5 % puts "Probs: |00>=[expr {[lindex $st 0]**2}] |11>=[expr {[lindex $st 3]**2}] others=0.0"
Probs: |00>=0.4999999999999999 |11>=0.4999999999999999 others=0.0
(tcl) 6 % puts "Correlation example: measuring qubit 0 as 0 forces qubit 1 to 0; as 1 forces 1."
Correlation example: measuring qubit 0 as 0 forces qubit 1 to 0; as 1 forces 1.
(tcl) 7 % # Interference: add amplitudes from two paths (phase sign +1 constructive, -1 destructive)
proc interfere {path1 path2 phase_sign} {
set total_amp [expr {$path1 + $phase_sign * $path2}]
set prob [expr {$total_amp ** 2}]
return [list $total_amp $prob]
}
puts "Interference Tests:"
set r [interfere [expr {1.0/sqrt(2)}] [expr {1.0/sqrt(2)}] 1.0]; puts "Constructive (phase 0): amp [lindex $r 0] prob [lindex $r 1]"
set r [interfere [expr {1.0/sqrt(2)}] [expr {1.0/sqrt(2)}] -1.0]; puts "Destructive (phase pi): amp [lindex $r 0] prob [lindex $r 1]"
set r [interfere 0.8 0.6 1.0]; puts "Mixed paths: amp [lindex $r 0] prob [lindex $r 1]"Constructive (phase 0): amp ~1.0 prob ~1.0
Destructive (phase pi): amp ~0.0 prob ~0.0
Mixed paths: amp 1.4 prob 1.96
(tcl) 3 % puts "Interference Tests:"
Interference Tests:
(tcl) 4 % set r [interfere [expr {1.0/sqrt(2)}] [expr {1.0/sqrt(2)}] 1.0]; puts "Constructive (phase 0): amp [lindex $r 0] prob [lindex $r 1]"
Constructive (phase 0): amp 1.414213562373095 prob 1.9999999999999996
(tcl) 5 % set r [interfere [expr {1.0/sqrt(2)}] [expr {1.0/sqrt(2)}] -1.0]; puts "Destructive (phase pi): amp [lindex $r 0] prob [lindex $r 1]"
Destructive (phase pi): amp 0.0 prob 0.0
(tcl) 6 % set r [interfere 0.8 0.6 1.0]; puts "Mixed paths: amp [lindex $r 0] prob [lindex $r 1]"
Mixed paths: amp 1.4 prob 1.9599999999999997This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks, gold 5Jan2026
gold 01/30/2026. Added categories, so can find message in Wiki.
gold 2/3/2025. Testing, encountered initial difficulty in saving work? Long code blocks with or unmatched wiki markup can sometimes confuse the Tcl Wiki formatting engine, especially if fences are not balanced or a line begins with markup it treats specially.
gold 2/4/2025. Added Automatic Dump of Examples, Using ActiveState. Added temp double hatch border, ##.
Please place any comments here with your wiki MONIKER and date, Thanks.gold 1/30/2026
Note. Testing computer methods and computer programs, maybe wrong numbers.
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